tag:blogger.com,1999:blog-65504596440529152182024-03-13T19:00:03.854-07:00Simply phy<i>If you can't explain it to a six-year-old, you don't understand it yourself.</i>
Einstein. Or Feynman. Probably neither.Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.comBlogger14125tag:blogger.com,1999:blog-6550459644052915218.post-378536643051678932016-02-14T03:03:00.001-08:002016-02-14T10:04:08.756-08:00Gravity waved! (Q&A)<div style="text-align: justify;">
<span style="font-family: inherit;">If you've been online even for a moment in the last few days, you might have heard something about it. Gravitational waves have been detected! The starting point to understand what those are is general relativity, and I <b>am </b>planning to write a post about it one day. But, due to popular demand,<sup>[<span style="color: blue;"><i>citation needed</i></span>]</sup> I will go ahead of myself and answer some questions about this exciting observation.
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<span style="font-family: inherit;">Note that there are many other sources from where you can learn about gravitational waves in general, and the measurement in particular. Just a few suggestions: the <a href="http://physics.aps.org/articles/v9/17?utm_source=email&utm_medium=email&utm_campaign=prl-ligo-2016">American Physical Society viewpoint</a> (the paper was published in Physical Review Letters, a scientific journal published by the APS), the <a href="http://www.nytimes.com/2016/02/12/science/ligo-gravitational-waves-black-holes-einstein.html?action=click&contentCollection=science&region=rank&module=package&version=highlights&contentPlacement=1&pgtype=sectionfront&_r=1">New York Times take on the story</a>, or simply <a href="https://www.youtube.com/watch?v=4GbWfNHtHRg&feature=player_embedded">this cool PhD comics video</a>. I hope that the Q&A below will further quench the thirst for understanding of the scientific enthusiast. </span></div>
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<span style="font-family: inherit;"><b>Q:</b> So is that just another type of a wave?</span><br />
<b>A:</b> This one is quite unique. Other types of waves, like waves in water, sound waves, or electromagnetic waves (light), are in some ways very different from one another, but they do share an important characteristic. Namely, they can be described as an oscillating modulation of something physical, be it the water particles, air pressure, or electric field intensity. Mathematically, we would write this as some amplitude that depends on position and time - <i>A(x, t)</i> - and the wave nature appears in the fact that there is some periodicity in the time dependence. Gravitational waves, however, are different and kinda hard to imagine properly, because it is the position and time intervals <b>themselves </b>that change! Mathematically, one might write the effect of a gravitational wave as something like <i>x'(x, t), t'(x, t)</i>, which is to say that space and time themselves change when such a wave passes through. (Note: this is not the way one would typically express the waves mathematically, but I think it's a good illustration.)</div>
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<b>Q: </b>So, like, a wave stretching the fabric of space and time?<br />
<b>A: </b>You hear that a lot these days, but I don't like the 'fabric' part of the statement. It implies that space and time are made out of something. As <a href="http://simplyphy.blogspot.ch/2015/08/some-stuff-einstein-actually-said.html">I've already discussed</a>, Einstein's view was rather that there are objective, absolute events, which appear to us imbued with the qualities of a location and a moment in time. Measuring these qualities defines our subjective space-time, and the laws of relativity tell us how to communicate with observers with different subjective space-times. In any case, whether space-time is physical or a figment of our perception is a big can of philosophical worms. I would thus much rather just say 'a wave stretching space-time' and throw the 'fabric' away to keep the interpretation load to a minimum.<br />
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<b>Q:</b> So what did we detect?<br />
<b>A: </b>General relativity explains gravity as curvature of space-time. Our Sun for example curves our space-time in such a way that if we try to follow a 'straight' path, we end up orbiting around it. This is a fairly static effect: mass implies curvature just by being somewhere. Another prediction of the theory, however, is that <b>waves</b> of space-time curvature can be radiated by <b>accelerating </b>massive objects. An extreme example of accelerating masses is a system of two black holes rotating around one another. This is what we detected. As black holes can pack a lot of mass in a tiny volume, the centrifugal acceleration in such a system is tremendous, and so is the amplitude of the emitted gravitational waves - at least as compared to any other emitters. Compared to the sensitivity of our apparata, the amplitude was still barely big enough for us to measure - and we've been trying for decades. In short, we detected two black holes that circled around one another and eventually collided 1.3 billion years ago.<br />
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<b>Q:</b> Ok, I know enough astronomy to know that we see something that happened 1.3 billion years ago because it happened 1.3 billion light years away from us. So did we, like, see two black holes about to collide somewhere (1.3 billion light years away), point our gravitational wave detector in that direction, and see the waves coming in? </div>
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<b>A: </b>The first part of the statement is correct: gravitational waves also travel at the speed of light, so indeed the black holes were 1.3 billion light years away. The second part is wrong. For starters, we cannot 'point' our gravitational wave detector in any particular direction: it's a giant device that 'listens' for space-time disturbances coming from anywhere (do see the <a href="https://www.youtube.com/watch?v=4GbWfNHtHRg&feature=player_embedded">PhD comics video</a>). More importantly, however, we had <b>no other </b>way to detect this black hole collision: we only know about it because of the LIGO measurement. In fact, even though the first-ever gravitational wave detection is what made the headlines, it is worth noting that this is also the first-ever detection of a black hole binary system, and of two black holes colliding to form one! </div>
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<b>Q:</b> Well that's pretty cool.<br />
<b>A: </b>Yes! A great deal of the excitement comes from the fact that we now have a whole new tool to explore the Universe with. Black holes in particular are very elusive; up until recently, we didn't even know whether they existed. It's easy for us to observe stars, since they emit so much light, but light cannot be emitted from the inside of a black hole. Thus, our observations are always based on indirect methods that typically rely on the extreme space-time bending in the vicinity of black holes. For example, light also gets bent, which results in the so-called <a href="https://en.wikipedia.org/wiki/Gravitational_lens">gravitational lensing</a> that we can observe and consequently infer the presence of a black hole. Gravitational-wave experiments will now add another dimension to our observations - and hence understanding - of the Universe.</div>
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<b>Q:</b> What did the collision look like?<br />
<b>A: </b>To us, it looked like this:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWaFKRTMBLw1bQuchjWKaqC9Hpg0Pn2tIF4Jw_N7SWGl2VQtuqHELeAg1tohpvaLe9qlC1qbVaJULonqUPM6-HwNulqhsq6aApHL59KD-uASjL3pxm0XmVjWoFpGD8NPa7ZHHyMZdmw0U/s1600/Picture1.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="355" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWaFKRTMBLw1bQuchjWKaqC9Hpg0Pn2tIF4Jw_N7SWGl2VQtuqHELeAg1tohpvaLe9qlC1qbVaJULonqUPM6-HwNulqhsq6aApHL59KD-uASjL3pxm0XmVjWoFpGD8NPa7ZHHyMZdmw0U/s1600/Picture1.png" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">B. P. Abbott et al. (LIGO Scientific Collaboration and the Virgo Collaboration), <i>Phys. Rev. Lett.</i> <b>116</b>, 061102 (2016)</td></tr>
</tbody></table>
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The top curve shows the signal that was recorded by the LIGO detector. The bottom curve shows our prediction (based on General Relativity) of what the signal from two colliding black holes would look like. The agreement between the two curves confirms the observation (careful statistical analysis was made to substantiate that claim). It is also worth noting that the same signal at the same time was detected by a similar detector located in Louisiana. This is important for us to be sure that it wasn't some fluke due to random noise. </div>
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<b>Q:</b> Isn't that a bit contrived? To me, that doesn't look anything like two black holes colliding...</div>
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<b>A:</b> Your brain does similar inferences all the time. Everything that you perceive as 'real' is your brain's interpretation of the signals it receives from your sensors, like your eyes. 'Seeing' something is an electromagnetic signal recorded by your retina not too different from the one in the figure above. It's your brain who is responsible for your subjective feeling of 'really' seeing it. We tend to think that we are <b>really</b> perceiving something when the interpreting process is sub-conscious. What we are doing with gravitational waves is, we are using our brains again, but this time consciously - through Einstein's theory. This is because they just don't have the capacity to record and interpret the gravitational-wave signal on their own. </div>
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<b>Q:</b> Does this change everything we thought we knew about the Universe?<br />
<b>A: </b>On the contrary, it confirms once again the extremely successful theory of General Relativity. Einstein predicted the existence of gravitational waves exactly 100 years ago, and, like a prophet who is actually worth listening to, he was once again correct. Actually, General Relativity works so well when it comes to astronomically-sized objects that I don't think there were many physicists doubting the existence of gravitational waves. For me at least, their existence was obvious, I just wasn't sure if I'd live to see a detection, since they are so elusive. It's pretty cool that I did. And of course it's very important to have the experimental confirmation, as we should never over-trust our theories. All in all, this is one more beautiful testimony to the fact that science <a href="https://xkcd.com/54/">just. works. bitches.</a> </div>
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<b>Q:</b> Is this quantum gravity?<br />
<b>A: </b>No, this is classical General Relativity stuff. We actually don't have a complete theory of quantum gravity due to the infamous difficulty of combining gravity with the other forces that are now explained by the Standard Model on the quantum level. A theory of quantum gravity might contain a <i>graviton</i> - a particle that 'carries' the gravitational interaction - but this is for now hypothetical. And since I'm <a href="http://simplyphy.blogspot.ch/2015/04/down-with-duality.html">not a big fan of the 'particle' notion</a>, I'd rather think of the graviton in relation to the gravitational waves: the graviton is the quantum of a gravitational wave, i.e. the smallest indivisible amount of energy that can be radiated by such a wave. There is a perfect analogy between this and a beam of light which is composed of photons - indivisible packets of light energy. In the case of a macroscopic number of quanta, or packets, the quantum nature is lost and both light and gravity can be described by the corresponding non-quantum theories (Maxwell's equations and General Relativity, respectively).<br />
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Still, it's worth noting that the LIGO experiment <b>also</b> managed to set an upper bound on the mass of a graviton, provided it exists.<br />
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<b>Q: </b>Thanks!<br />
<b>A: </b>No prob. </div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-53265844271920848462016-01-18T05:30:00.001-08:002016-01-18T05:30:47.498-08:00Let me teal you about the cyansce of color<div class="MsoNormal" style="text-align: justify;">
If you were among the asocial few who actually listened to their Physics teacher, you'll probably know most of what follows. But maybe you were cool instead, and even if not, surely some of the details go beyond what you know. In any case, this is one more post (after <a href="http://simplyphy.blogspot.bg/2015/05/energy-is-great-name-for-band.html">this one</a>) on the way of explaining what I did during my PhD. Enjoy. <br />
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<span style="font-family: inherit;">What gives objects color?<o:p></o:p></span></div>
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<span style="font-family: inherit;">Color, of course, has everything to do with light. Now, light is actually electromagnetic radiation, and as such propagates in waves, and every wave has an
associated wavelength. Classically, this is the distance between two consecutive crests. Quantum mechanically, the wavelength also determines the
smallest amount of energy that a beam of light can carry - namely, the energy
of one light 'quantum', commonly called a <i>photon</i>. The higher the
wavelength, the lower the photon energy. I’ll get back to this later on. Importantly, individual wavelengths, or photon energies, are <b>also</b> associated to
various colors. Or, rather, the other way round - to every color, a particular
wavelength can be associated, but not every wavelength has an associated
color. This is simply due to the limitation of our perception - we cannot register all the wavelengths, and the concept of color is only associated to the ones we do. You should be familiar with
some version of the following image, which illustrates the full range of possible wavelengths, and what we call the corresponding radiation:</span></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNKNdcI85X8izr7u4fommM08cN0ruXP8xvDcDM7QRLofACbnBF6YLj-A_LDzcVOM6An6QOAX7y9_N3Ruxj2WmQBu46NxMRCOXhxwmmiaqXTaP8Zc2uO_5eSkZy0qJG3HcjRfleHavxslY/s1600/593px-Electromagnetic-Spectrum.svg.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="611" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNKNdcI85X8izr7u4fommM08cN0ruXP8xvDcDM7QRLofACbnBF6YLj-A_LDzcVOM6An6QOAX7y9_N3Ruxj2WmQBu46NxMRCOXhxwmmiaqXTaP8Zc2uO_5eSkZy0qJG3HcjRfleHavxslY/s1600/593px-Electromagnetic-Spectrum.svg.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">"Electromagnetic-Spectrum" by Victor Blacus. Licensed under CC BY-SA 3.0 via Commons - https://commons.wikimedia.org/wiki/File:Electromagnetic-Spectrum.svg#/media/File:Electromagnetic-Spectrum.svg</td></tr>
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As you can see, there is only a small range of wavelengths that we can see, namely the one marked as 'visible light'. Now, a beam of light of a fixed wavelength is called <i>monochromatic
</i>light, and indeed it has a fixed color (if it is within the visible range).
What is important to understand, however, is that light around us is virtually
never of a single wavelength - instead, it has contributions of varying
intensity at <b>all</b> possible wavelengths. The intensity of the light as a
function of its wavelength is what is called the <i>spectrum</i>. Color is
almost the same thing, but it requires in addition a brain to recognize the
spectrum and give it the color-label. For color is certainly in the eye of
the beholder, and the way two identical spectra
are recognized differs among different animals, and even among different people.
Here, I won't be discussing the human eye; instead, let's just assume a generic observer that registers equally well light
of wavelengths from 400nm to 700nm, see image above (nm stands for <i>nanometer</i>, which is
one billionth of a meter). Anything outside of this <i>spectral range</i> is,
in our discussion here, irrelevant. In other words, the following two light
sources appear to be of the same color to a human, since they have the exact
same intensity vs. wavelength dependence in the spectral range that we
register.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiagRW2MvG8WZheUPOFP99KHVq2oHO25Sko1xKfEIsehqWHK3a4G1WfhKDrlP-pXt-299W03zfWQdqvptPOa41ExTSC-DC_qmt0TnSp-un_akfmhkHb-GSM1Vwq4P0TZ1vFgSztIZ9UGlE/s1600/spectrum12.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="369" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiagRW2MvG8WZheUPOFP99KHVq2oHO25Sko1xKfEIsehqWHK3a4G1WfhKDrlP-pXt-299W03zfWQdqvptPOa41ExTSC-DC_qmt0TnSp-un_akfmhkHb-GSM1Vwq4P0TZ1vFgSztIZ9UGlE/s1600/spectrum12.png" width="480" /></a></div>
In short, the color of a beam of light is determined by the
contributions of the various colors - or wavelengths - that make it up. In the
example above, the intensity is the strongest in the blue region, and so we would
probably register both of the spectra as blue-ish. However, in general, all the
contributions mix up - proportionally to their intensity - to determine the
final color. Much like in an artist’s palette, or in primary school art lessons.</div>
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Now, the range of wavelengths that our eyes detect is not at all
arbitrary. This is what the spectrum associated to sunlight reaching the Earth
looks like:<o:p></o:p></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6Z43dz11U7y6Lag4FLh6Ahjf7__mAIzosYFGOd0NiRGs2eTjo1RqRjd7AYQV16EnotTnqfelneBnnMDLZ4swYcO2WmfNKQ7-IQT3slxi5b2HehtVcFhYVsNLWchxKvwcFmxAPiVXWkq8/s1600/640px-Solar_spectrum_en.svg.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6Z43dz11U7y6Lag4FLh6Ahjf7__mAIzosYFGOd0NiRGs2eTjo1RqRjd7AYQV16EnotTnqfelneBnnMDLZ4swYcO2WmfNKQ7-IQT3slxi5b2HehtVcFhYVsNLWchxKvwcFmxAPiVXWkq8/s1600/640px-Solar_spectrum_en.svg.png" width="480" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">"Solar spectrum en" by Nick84. Licensed under CC BY-SA 3.0 via Commons - https://commons.wikimedia.org/wiki/File:Solar_spectrum_en.svg#/media/File:Solar_spectrum_en.svg</td></tr>
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Our eyes have naturally evolved to detect light in the range
in which there is the most of it! If it’s not clear why, I’d have to explain
natural selection to you, and that’s well beyond this post. But sunlight brings
me to the next point: so far we only discussed color in
relation to the spectrum of a beam of light. With this in mind, the color of objects is
naturally determined by the light coming from them. However, since most objects around us don’t emit light on their own - at least not in the visible range - color is implicitly also determined by the spectrum of the light <i>illuminating </i>them. In
other words, if we were to illuminate any object with monochromatic blue light,
it would appear either blue or black, or something in between - because no
other hue can mix up in the spectrum. Thus, in a sense, we should call the 'true' color of an object the one we see after illumination with light
of equal intensity at all wavelengths, otherwise known as <i>white light</i>. Sunlight is to a good approximation white light, see image above.</div>
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Finally, I come to the main point. What determines the color
of an object once we <b>fix </b>the observer (an average human) and we <b>fix </b>the
illumination (white light)? Both of these factors are important, but they are <i>extrinsic
</i>to the object itself. There must also be some <i>intrinsic </i>property
that gives an object its hue. Indeed, its physical and/or chemical structure determine its optical features, and more specifically -
the absorption, reflection, and transmission properties. These are related to a
wide variety of phenomena and can be incredibly complex to study and predict, but
here we take them for granted for illustrative purposes. The important point is
that there are several possible outcomes for light hitting an object, and the
interplay between those determines its color. Namely, light can be transmitted,
reflected, scattered, or absorbed, and normally all of those happen
simultaneously - to a various extent.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhMUuHEamMDW1wicb8SADsvzQ2DwiwzridvnN6JdlIxtsNQGHq4_ZDcWBoFYjaLe9DnPFXVkwEgAEZV1HEahj-o8iq4xopX4qUxrrOoP_Vyp36GzQVR7HFSHsnPMK9lAcw3uYGx8FupH0/s1600/optical_properties.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="181" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhMUuHEamMDW1wicb8SADsvzQ2DwiwzridvnN6JdlIxtsNQGHq4_ZDcWBoFYjaLe9DnPFXVkwEgAEZV1HEahj-o8iq4xopX4qUxrrOoP_Vyp36GzQVR7HFSHsnPMK9lAcw3uYGx8FupH0/s1600/optical_properties.png" width="400" /></a></div>
It is in principle possible that the object itself emits
some light, which adds to or even dominates its color. However, as mentioned earlier, the color of the vast majority of everyday stuff is determined by the four properties illustrated above, and not by emission. Actually, all objects around us, including ourselves, <b>do </b>emit
electromagnetic radiation, but the intensity is peaked in the infra-red, making it invisible to us. The peak of the emission spectrum is
determined by the temperature of the object. This is why incandescent light-bulbs, for
example, in which the wire becomes much hotter than room temperature, do emit
light at wavelengths that we see. But, again, we neglect emission in our discussion, and focus on illumination. Here are
several basic examples of colors and how they arise from the interplay of
transmission, reflection, scattering, and absorption. Black bodies absorb most of the incident light:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJuIxm7u-b-u2eZRY11kOJaU39gHU_XoKpjZZcAcpsIrNk1gfgXfEJSFL9nNEKuHllPhyJ-B0iDNL5SeLQvYTZ4yAfqFlRlAenuthSYm28LljitGRgIW1xjT0poJrzWYd8mQZkyZI3ujw/s1600/four_black.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJuIxm7u-b-u2eZRY11kOJaU39gHU_XoKpjZZcAcpsIrNk1gfgXfEJSFL9nNEKuHllPhyJ-B0iDNL5SeLQvYTZ4yAfqFlRlAenuthSYm28LljitGRgIW1xjT0poJrzWYd8mQZkyZI3ujw/s1600/four_black.png" width="400" /></a></div>
And white things scatter most of the light back.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq32Nqx90jVS6yobj4AIu3No2io1wUGhYDlzzlwJ7p-pvM8KeLTNhkhvq_EDayYgcG7jbdfgBS1zKAmdaIAaSeV1Zxkc4k1KkkWfa8GSINtr4teGGapKg7SN9UCxBGvwClYatbIHR2CBo/s1600/four_white.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq32Nqx90jVS6yobj4AIu3No2io1wUGhYDlzzlwJ7p-pvM8KeLTNhkhvq_EDayYgcG7jbdfgBS1zKAmdaIAaSeV1Zxkc4k1KkkWfa8GSINtr4teGGapKg7SN9UCxBGvwClYatbIHR2CBo/s1600/four_white.png" width="400" /></a></div>
Here, a distinction should be made between scattering and reflection. When the reflection dominates, the object is still technically white. However, we would be more prone to calling this object a mirror, because we'd be more used to seeing the exact same color of light that hits it. Still, if you think about looking at a mirror from far away when it's in the sunlight, it looks like a bright white object. The difference is that the reflected light is directional, and thus far stronger in the reflection direction than the scattered light, which goes everywhere. Have a look at <a href="https://www.youtube.com/watch?v=-yrZpTHBEss">this Vsauce video</a> for further insight into the color of mirrors.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsS20oW0ZtVbjyp8aDV7XgaxoUsmw_E9aRvmdMDYIRzWe3brKoZj8hEXSBoO4k-BAgpVsUBGo7B5IPIZdUy5Z1jTrXZKVR_mXsi_4zhxerJWNED92UoAFugcGLGWwm1ZEHKtsoAmTUR_k/s1600/four_mirror.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsS20oW0ZtVbjyp8aDV7XgaxoUsmw_E9aRvmdMDYIRzWe3brKoZj8hEXSBoO4k-BAgpVsUBGo7B5IPIZdUy5Z1jTrXZKVR_mXsi_4zhxerJWNED92UoAFugcGLGWwm1ZEHKtsoAmTUR_k/s1600/four_mirror.png" width="400" /></a></div>
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Finally, when there is mostly transmission, the object appears translucent because we see the light coming from behind rather than the illuminating light. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilcT4_VhJr21VYTUUpfqxzHi5cfIrDoxHG3C_SmrKPHdhVJkYkvci02W42QWXvVjgMj0KQchcvj0mW9H6CJLoKny3Jno06XQCH-0N6udvDi7x1FAzLdxEyz1oWjkYXH11crW2cD8Ne6-M/s1600/four_transparent.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilcT4_VhJr21VYTUUpfqxzHi5cfIrDoxHG3C_SmrKPHdhVJkYkvci02W42QWXvVjgMj0KQchcvj0mW9H6CJLoKny3Jno06XQCH-0N6udvDi7x1FAzLdxEyz1oWjkYXH11crW2cD8Ne6-M/s1600/four_transparent.png" width="400" /></a></div>
To make matters more complex still, color depends on the interplay between these four phenomena not just in general, but at <b>every </b>wavelength within the visible range. Thus, for example, an object could be mostly absorbing at all wavelengths apart from blue light, which gets mostly scattered. Such an object would appear blue.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg206Uud92HVZhAPl8bObrfYYqUgq2VYGXOQr9iMVqGtXMX1ud0t85X2ZzocI2Pevpz5V1Ft03sYrcU6NO-YhyvrMM-Gg-FhEf37EeKlOjVLx_3WiyFYeoyrPsDa7qP7Eef3-oKGDjFcuw/s1600/four_blue.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg206Uud92HVZhAPl8bObrfYYqUgq2VYGXOQr9iMVqGtXMX1ud0t85X2ZzocI2Pevpz5V1Ft03sYrcU6NO-YhyvrMM-Gg-FhEf37EeKlOjVLx_3WiyFYeoyrPsDa7qP7Eef3-oKGDjFcuw/s1600/four_blue.png" width="400" /></a></div>
And, as for white objects above, if the blue light were mostly reflected rather than scattered, the color would be brighter, glossier blue, with some reflections of nearby objects appearing (in blue). Think car paint.<br />
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These qualitative consideration give a lot of intuition about color. So, while we are at it, why not answer the age-old mystery:
why is the sky blue? In fact, it's not really - or rather, it can take on various colors, blue being just one of them. The first important note is that the sky is actually mostly translucent (assuming no clouds). When you look at the Sun, you see it (right?), and when you look back at Earth from space, you see it. Furthermore, the Sun looks white (technically kinda yellow-ish because the spectrum is not exactly white light), and around the Sun the sky looks white, too. It doesn't look blue, because we're mostly seeing the light coming directly from our star.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfks7eSgNah-lj2NqxZU0ND28_E3p4KF9FR6pKvpuMskzC_eYjKaEL8PxuLuygDjb0ROK7kY6Sgd55e2tbbp3LDgM2MKuYT4SsDMH5EVvFMRp0MxWLwzORIT9xdfBeEFGkAa3g7GBBmbE/s1600/sky_white.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="338" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfks7eSgNah-lj2NqxZU0ND28_E3p4KF9FR6pKvpuMskzC_eYjKaEL8PxuLuygDjb0ROK7kY6Sgd55e2tbbp3LDgM2MKuYT4SsDMH5EVvFMRp0MxWLwzORIT9xdfBeEFGkAa3g7GBBmbE/s1600/sky_white.png" width="400" /></a></div>
You can have a quick look to make sure that's the case, but, needless to say (I hope), please be careful with burning your retinas out. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyDuhc1hTEjanrB9kuE_cYBlayYBAdQTnXZwPTDeWi7-eVrCP7qkSPJ0oj6I4Kp47c3Kudj5MA4WHGgCPuMwYbGLBeTseUREQ4f5obuOPrUT1gjXkzRp_SMwgIHk20DE3AeMVzlH0a29w/s1600/sky_black.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="338" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyDuhc1hTEjanrB9kuE_cYBlayYBAdQTnXZwPTDeWi7-eVrCP7qkSPJ0oj6I4Kp47c3Kudj5MA4WHGgCPuMwYbGLBeTseUREQ4f5obuOPrUT1gjXkzRp_SMwgIHk20DE3AeMVzlH0a29w/s1600/sky_black.png" width="400" /></a></div>
Apart from that, the sky is indeed blue when we look away from the Sun, at the light that is scattered by the constituent particles of our atmosphere. Now, it turns out that light of a given wavelength gets scattered stronger the lower the wavelength. This means that light in the blue-violet part of the spectrum gets scattered the most (in the visible range). And that's why we see a blue color when we look away from the Sun.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRKRawb9tCawPShNJq1gm57AoL27xipqLcfaT8ZXQ8DvAfVEiQ9-FEpj1OpN5RBELO153zTIdQ5WBrj4YkKITO1dRNI6RTXHAcHDytIF2cUPcS3kggsYeYcYK8H0s-n1CoUZWJOd2pUmE/s1600/sky_blue.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="338" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRKRawb9tCawPShNJq1gm57AoL27xipqLcfaT8ZXQ8DvAfVEiQ9-FEpj1OpN5RBELO153zTIdQ5WBrj4YkKITO1dRNI6RTXHAcHDytIF2cUPcS3kggsYeYcYK8H0s-n1CoUZWJOd2pUmE/s1600/sky_blue.png" width="400" /></a></div>
Of course, that's not the full story. We also all know that the sky appears red at sunset and sunrise. The explanation is pretty much the same, only now sunlight travels a long distance through the atmosphere. Since red is on the long-wavelength part of the spectrum, the red component of the light is the last to scatter out before reaching us. Thus, now, when looking directly at the sun, it appears red-ish, as does the region around it.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmLCz2_XH_efJkWNK7SGVpIpxs9AZnLiAD7D-CyQF9L7_XorQrajsEPdpQ2BTVwTBQm2PGyS3xy_euglBW7vPPv51Ym9JPc4v2JPGbzcrZWaPcIaW3MfO1kxLloaxAPqIMPfETODzSu58/s1600/sunset_red.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmLCz2_XH_efJkWNK7SGVpIpxs9AZnLiAD7D-CyQF9L7_XorQrajsEPdpQ2BTVwTBQm2PGyS3xy_euglBW7vPPv51Ym9JPc4v2JPGbzcrZWaPcIaW3MfO1kxLloaxAPqIMPfETODzSu58/s1600/sunset_red.png" width="480" /></a></div>
On the other hand, if we look away from our star, the sky is still blue-ish, or white-ish, or just getting dark, because most of the light gets scattered away before reaching us.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-cE9b-aMVPO0QxMupkIXwMxazZei6A3HC-LSw-gRxV9c5RVg4yCjhhPrkoQDsil89e-_KiaNPdkjWr3FtxQrfU_S-YM9BPHEN04PrK5WQJM6282T8XmnTG5EVBAfTYtcZ1KddZMTz9sM/s1600/sunset_blue.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-cE9b-aMVPO0QxMupkIXwMxazZei6A3HC-LSw-gRxV9c5RVg4yCjhhPrkoQDsil89e-_KiaNPdkjWr3FtxQrfU_S-YM9BPHEN04PrK5WQJM6282T8XmnTG5EVBAfTYtcZ1KddZMTz9sM/s1600/sunset_blue.png" width="480" /></a></div>
All this constitutes the well understood, and maybe boring, part of the story. When physiology and psychology kick in, it gets much, much messier, as color perception gets very subjective. Note for example that <a href="http://www.apa.org/monitor/feb05/hues.aspx">language shapes the way we perceive colors</a>. Blue is a particularly weird color, and believe it or not <a href="http://www.theguardian.com/books/2010/jun/12/language-glass-colour-guy-deutscher">there's evidence</a> that you might not know that the sky was blue had you not heard that all your life. But, at least when it comes to Physics, you now know why it should be somewhat blue-ish. Apart from the cases when it's white-ish or red-ish... </div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-41165211163105942482015-09-21T09:42:00.000-07:002015-09-21T09:42:42.737-07:00Wibbly-wobbly timey-wimey... stuff<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><span style="font-family: inherit;">Disclaimer: this post quickly becomes quite technical, but it could give some extra STR
intuition to those brave enough to stick with it. <o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;"><br /></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">I'll try to explain
relativity geometrically. This is actually a very common approach, because space and
time themselves are best understood geometrically. As discussed in the '<a href="http://simplyphy.blogspot.ch/2015/08/some-stuff-einstein-actually-said.html">introduction</a>', these notions are
intricately related to our <b>experience</b>:
our perceptions tell us that things can be in different places and can happen
at different moments, the collection of which we call space-time. Very often,
the need arises to quantify the places and moments in order for us to, you know, function
as humans and as a society. In our day-to-day lives, we go about this in an informal way, choosing just the right amount of vagueness that is needed for particular situation. </span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHstSJC_NhqFsVrZHNcL-4Yy-2FESuMaK7CEyOv7fkYRAEMM272f7cdi2YkijXe6igkxtrfyhFsFnoNYctEOapywX2KmzLoE5GuSoiAVyS-NGfReVtR6KanGHjKSdZBpTGkAnuMbqFSPc/s1600/store.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="329" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHstSJC_NhqFsVrZHNcL-4Yy-2FESuMaK7CEyOv7fkYRAEMM272f7cdi2YkijXe6igkxtrfyhFsFnoNYctEOapywX2KmzLoE5GuSoiAVyS-NGfReVtR6KanGHjKSdZBpTGkAnuMbqFSPc/s1600/store.png" title="I do expect a smartphone version of this coming up" width="500" /></a></span></span></div>
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<span style="font-family: inherit; line-height: 115%;">The formal (i.e. boring?), scientific way to describe positions is to use a system of <i>coordinates</i>, and the most commonly used one is <i>Cartesian coordinates</i>,
introduced, </span><span style="font-family: inherit; line-height: 18.4px;">just like <span id="goog_2111377107"></span><a href="https://en.wikipedia.org/wiki/Evil_demon">the evil demo<span id="goog_2111377108"></span>n</a>,</span><span style="font-family: inherit; line-height: 115%;"> by </span><span style="font-family: inherit;">René</span><span style="font-family: inherit; line-height: 115%;"> Descartes. In two
dimensions, these look in this probably familiar way:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmA_rGzFDM90ZGvY6Lah0uMidJoP3t-V8ieEH0MxCPbicpnTEimdxMYLf6-ei3avGl4qwZnIWwBEUjhRQSr6E3h51D4VR5JOeF3GhSvGpBy9KMFvGAlepwC2NQ6o4ch2TzGu_7jxOwSuM/s1600/cart_coord.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="183" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmA_rGzFDM90ZGvY6Lah0uMidJoP3t-V8ieEH0MxCPbicpnTEimdxMYLf6-ei3avGl4qwZnIWwBEUjhRQSr6E3h51D4VR5JOeF3GhSvGpBy9KMFvGAlepwC2NQ6o4ch2TzGu_7jxOwSuM/s200/cart_coord.png" width="200" /></a></div>
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<span style="font-family: inherit; line-height: 115%;">What I mean by </span><i style="font-family: inherit; line-height: 115%;">two dimensions</i><span style="font-family: inherit; line-height: 115%;"> is that the <i>space </i>described by the example above (e.g. the plane of your monitor) is such that any <i>location</i>
can be precisely pinpointed by two numbers. These could be the x- and
y-positions in the Cartesian frame above, but the concept of
dimensionality is much more general. In fact, we are intuitively used to a very complex two-dimensional space, since we are tied by gravity to the surface of the Earth <a href="https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica">[1]</a>. This surface has an extremely complicated shape, which the x-y plane above cannot capture well, but it is still two-dimensional, in the sense that any location can be uniquely defined by two
numbers, as anybody who's used GPS coordinates knows. So 3D beings as we are, we do anyway spend our lives tied down to what is, to a good approximation, a 2D world. </span></div>
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<span style="font-family: inherit; line-height: 115%;"><br /></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">The Cartesian coordinate
system seems very natural. Right angles make sense. Well, I don’t know if it is a priori aesthetic - in fact I think that <i>a priori aesthetic </i>is an oxymoron - but we humans definitely seem to like it a lot. We even like to organize our cities in this way. There's not much difference between saying '42nd and 5th' and saying 'at <i>x = 3.2</i>, <i>y = 2</i>', or something.</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmHWbx11zh2suwfxd_pL6DOABgoj0d3BApUwulsTlSZWIa8qDnYuxfzF2zfRzrcgqen9pmLKqOJro6nCSwlN7r5UcTjp9wx_pAYN_5Ov65yY_GX1QTSGGatHcbVsbBFdv4DdZ9RfjUW3o/s1600/nyc_xy.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmHWbx11zh2suwfxd_pL6DOABgoj0d3BApUwulsTlSZWIa8qDnYuxfzF2zfRzrcgqen9pmLKqOJro6nCSwlN7r5UcTjp9wx_pAYN_5Ov65yY_GX1QTSGGatHcbVsbBFdv4DdZ9RfjUW3o/s1600/nyc_xy.png" width="400" /></a></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">Now, to illustrate
relativity, we're going to decrease the spatial dimensions to one. That means that we’re only going to keep the x-axis. It also means that anything
living in this toy 1D world can only be in front or behind anything else. The reason for this simplification is
because we want to use the other axis to represent time:<o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeIdONoADB6JqTOiN_OIgjYLWvFSvwAYSIIffhJlooe3PX6AxkBhEe1zpL3sTGrNz4GDNDtdf93yNX2gMlwT5nsw5qw0XvzXLhbipE4TXmtao1H78CwVUGPUi9Ub6Go_LTKNOFPG82QRo/s1600/xt_coord.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="183" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeIdONoADB6JqTOiN_OIgjYLWvFSvwAYSIIffhJlooe3PX6AxkBhEe1zpL3sTGrNz4GDNDtdf93yNX2gMlwT5nsw5qw0XvzXLhbipE4TXmtao1H78CwVUGPUi9Ub6Go_LTKNOFPG82QRo/s200/xt_coord.png" width="200" /></a></div>
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<span style="font-family: inherit; line-height: 115%;">This is now a graphical
representation of space-time, and already hints at the fact that those are
kinda like different aspects of the same thing. Note that we always had some intuition that time is very similar to space, or in any case certainly before Einstein or
Descartes. This can be seen in our language. Interestingly, there are different ways in which <a href="http://www.businessinsider.com/how-different-cultures-understand-time-2014-5?IR=T">different cultures view time</a>. What is practically ubiquitous among cultures, however, is the fact that the words we use to speak about time are very similar to the ones we use for space: there is a <a href="https://en.wikipedia.org/wiki/Conceptual_metaphor">conceptual metaphor</a> of time as a path through space. The examples are numerous. Plenty of time <i>ahead</i> of us. <i>At </i>8am. In French one even says, 'le moment </span><span style="line-height: 18.4px;">où</span><span style="font-family: inherit; line-height: 115%;"> quelque chose </span><span style="line-height: 18.4px;">s'est passé</span><span style="font-family: inherit; line-height: 115%;">', which literally means 'the moment <i>where</i> something happened.' </span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;"><br /></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">An important remark is due. The space-time <i>reference frame </i>shown above is neither unique nor absolute. It is in fact observer-dependent, so
you should think of everyone as having one of those attached to them.<o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_02fJGMOkMCnV1GGTiIzUAMMbNNU55QZLwYoNNUHiSLO7GhYdvA46b-1i1HizHDyvjy3Cw8JZMcmmQfwwGyAW6cnA8XfTCr3ckjzxrJUfLPaT-sOPTpZuGhgjR5xtxYq6eu0UkXX7odc/s1600/onmyface.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_02fJGMOkMCnV1GGTiIzUAMMbNNU55QZLwYoNNUHiSLO7GhYdvA46b-1i1HizHDyvjy3Cw8JZMcmmQfwwGyAW6cnA8XfTCr3ckjzxrJUfLPaT-sOPTpZuGhgjR5xtxYq6eu0UkXX7odc/s1600/onmyface.png" width="320" /></a></div>
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<span style="font-family: inherit; line-height: 115%;">The Universe itself
consists of </span><i style="font-family: inherit; line-height: 115%;">events </i><span style="font-family: inherit; line-height: 115%;">which <b>are</b> absolute
in the sense I discussed <a href="http://simplyphy.blogspot.ch/2015/08/some-stuff-einstein-actually-said.html">here</a>: all observers agree on them happening, but, to do that, every observer relies on their own reference frame. This is what this looks like.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOHES_a3R8aAFg54WD561OJBMoNpLtC8yOSeoz11ax5CYVHATWCB5mvNFOj8F8FcKJzee-QpHvO5sNch1SjqkpSJiqalH_8LNih9Y6YU25nuQ3NwH1XwJdN6AGvq_5rES3y-6RpKOOI_o/s1600/agreeing1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="396" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOHES_a3R8aAFg54WD561OJBMoNpLtC8yOSeoz11ax5CYVHATWCB5mvNFOj8F8FcKJzee-QpHvO5sNch1SjqkpSJiqalH_8LNih9Y6YU25nuQ3NwH1XwJdN6AGvq_5rES3y-6RpKOOI_o/s1600/agreeing1.png" title="If only internet arguments went equally smoothly" width="500" /></a></div>
<span style="font-family: inherit; line-height: 115%;">This way of talking about
events by naming both their spatial and temporal position is perhaps a bit awkward, but
you’d better get used to it if you want to understand relativity. Space-time
should always be thought of as indivisible.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCrWDYSpyU7woCIoWPW9Luhgsm2HLKV81mRUL3_4WDbSWEJDvCbCkxUvlnZZ4MrEbdJyOrKO7H1ZRu16iU3rNvKXsK0EVroL-klBOk-Bd7uz0VsLReANK6nh7SzMejeXDpBw8bFSw4LGY/s1600/theshow.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCrWDYSpyU7woCIoWPW9Luhgsm2HLKV81mRUL3_4WDbSWEJDvCbCkxUvlnZZ4MrEbdJyOrKO7H1ZRu16iU3rNvKXsK0EVroL-klBOk-Bd7uz0VsLReANK6nh7SzMejeXDpBw8bFSw4LGY/s1600/theshow.png" title="It's weird but practical to talk like that " width="400" /></a></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">The example above, where two
observers at a different place compare their observations of an event is fairly
straightforward to understand. It's also possible to compare reference frames at different <b>times</b>. This is slightly less intuitive, but still nothing too complicated.<o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEgyc2lgWMKyU5Hg-GB8G0RRA945_WFiRcxz9Rxt2BvPLJEYmuii6f5XmqhONT4ealNJbYFp5gI8binp9y3yaHyUsruFrdpI-iDA9S-3Z5zz106KNmenpkCLetMfyhNMLoSQs4jWjUdlM/s1600/agreeing2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="373" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEgyc2lgWMKyU5Hg-GB8G0RRA945_WFiRcxz9Rxt2BvPLJEYmuii6f5XmqhONT4ealNJbYFp5gI8binp9y3yaHyUsruFrdpI-iDA9S-3Z5zz106KNmenpkCLetMfyhNMLoSQs4jWjUdlM/s640/agreeing2.png" title="Also, is event A my eye?!" width="500" /></a></div>
<span style="font-family: inherit; line-height: 115%;">In both cases, to see whether the observers agree, they simply have to properly add or subtract the distance between them (both in space and in time) from their corresponding observations. This </span><i style="font-family: inherit; line-height: 115%;">transformation </i><span style="font-family: inherit; line-height: 115%;">of the reference frames is mathematically known as a </span><i style="font-family: inherit; line-height: 115%;">translation</i><span style="font-family: inherit; line-height: 115%;"> (technically, not related to language, but somehow this connotation is also appropriate here). All this was well understood already in pre-relativity physics. The main
innovation of STR is what happens to the reference frames of observers that
are </span><span style="font-family: inherit; line-height: 115%;"><b>moving</b></span><span style="font-family: inherit; line-height: 115%;"> with respect to one
another. This is what we used to think the transformation looked like:</span><br />
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<span style="font-family: inherit; line-height: 115%;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisBC5LfqazJin7WDcMTIm4VtkXaVnQu6lu7CIQubCir7TeI6CJkT6YCMYtxWWlS1CzVfU18blPgCh3RKAz_EUFJhD_FfycX_7dyhrgdAVSZE55mbMJ7cZk9UBWO_RdZZ91JIOSUMUqcW0/s1600/galilean.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisBC5LfqazJin7WDcMTIm4VtkXaVnQu6lu7CIQubCir7TeI6CJkT6YCMYtxWWlS1CzVfU18blPgCh3RKAz_EUFJhD_FfycX_7dyhrgdAVSZE55mbMJ7cZk9UBWO_RdZZ91JIOSUMUqcW0/s1600/galilean.png" width="400" /></a></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">The two reference frames
above correspond to two observers moving with respect to one another, which at<i> t = 0</i>, <i>x = 0</i> (in both frames) find themselves at the same time and place. This is a graphical representation of what is known as a <i>Galilean transformation</i>. The fact that the time axis tilts while the space axis doesn't is
deeply connected to our (wrong) understanding of time as something that is
absolute for everybody. Consider this. If we have two events which happen at
the same time with respect to observer one, then they also happen at the same
time for observer two. </span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwrxNxtdPcsQqQWAJLcMdxFosRWJb9IQAWSDG5SjXTjRlz-YqQNvaVxiZA8NX58HXat2tYNyQphJ7rVEWdtucyLfHzOTnxKBjwm-sMP7SRzq0vwK1IQDo7Q5w6DWVhyphenhyphenzhtn-wsonEzgmU/s1600/tatb.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="216" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwrxNxtdPcsQqQWAJLcMdxFosRWJb9IQAWSDG5SjXTjRlz-YqQNvaVxiZA8NX58HXat2tYNyQphJ7rVEWdtucyLfHzOTnxKBjwm-sMP7SRzq0vwK1IQDo7Q5w6DWVhyphenhyphenzhtn-wsonEzgmU/s1600/tatb.png" width="320" /></a></div>
<span style="font-family: inherit; line-height: 115%;">This shared </span><i style="font-family: inherit; line-height: 115%;">simultaneity</i><span style="font-family: inherit; line-height: 115%;"> of events means that we can come up with a common time-counting scheme for the two observers, such that </span><i style="font-family: inherit; line-height: 115%;">t = t' </i><span style="font-family: inherit; line-height: 115%;">for <b>all</b> events</span><i style="font-family: inherit; line-height: 115%;">. </i><span style="font-family: inherit; line-height: 115%;">This would represent the absolute-ness of time.</span><i style="font-family: inherit; line-height: 115%;"> </i><span style="font-family: inherit; line-height: 115%;">However, if two events happen at the same </span><b style="font-family: inherit; line-height: 115%;">place </b><span style="font-family: inherit; line-height: 115%;">for observer one, they don’t
happen at the same place for the second guy: space is relative. </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZpqUhFMaSRo7YRssXMR7SdDuZ_u5RFN3Cz-UoaKVcrj3DhsUK8PUkaxY_xOv2bTU4qGiWOfJvUicVWLwL1vo_YZJiUDamu5_TMhyphenhyphendEDrLmOJqujEiTfdO-Yzgx40JutFcu_el1A9hz1Q/s1600/xaxb.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="263" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZpqUhFMaSRo7YRssXMR7SdDuZ_u5RFN3Cz-UoaKVcrj3DhsUK8PUkaxY_xOv2bTU4qGiWOfJvUicVWLwL1vo_YZJiUDamu5_TMhyphenhyphendEDrLmOJqujEiTfdO-Yzgx40JutFcu_el1A9hz1Q/s1600/xaxb.png" width="320" /></a></div>
<span style="font-family: inherit; line-height: 115%;">To see what the situation
looks like in relativistic physics, let’s first take care of a small
technicality. In measuring and plotting <i>x</i> and <i>t</i>, we have to choose some <i>units</i>. We actually have some freedom in that, and the units we choose affect the </span><i style="font-family: inherit; line-height: 115%;">scale</i><span style="font-family: inherit; line-height: 115%;">
of the axes. Let’s say we take the standard choice of measuring time in
seconds. A standard choice for measuring positions is the metre, unless you have the misfortune to come from <a href="https://en.wikipedia.org/wiki/Metric_system#/media/File:Metric_system_adoption_map.svg">one of the three countries in the world</a> where it isn't. For the
purposes of relativity, however, it is particularly convenient to choose a different
unit: the light-second. This is analogous to the light-year (which we would use if we chose to measure time in years instead of seconds) in that
it’s defined as the distance that light travels for one second. With this choice of units, the
reference frames for observers moving with respect to one another turn out to look like this:</span><br />
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<span style="font-family: inherit; line-height: 115%;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNGvGGu-d9SbQhzXKDHvunDSj7DCGGkA8GfOOVOGtzQIuMQHfpVkykHNPSMYoaLAyJiEi_ARMIOPQLDXJaf1dU1ZLxyJy3mZk2dJ_fK4r6KKo36OyYXyJShc-D2XHcpDDp1Q7NGayU_uw/s1600/relativistic.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNGvGGu-d9SbQhzXKDHvunDSj7DCGGkA8GfOOVOGtzQIuMQHfpVkykHNPSMYoaLAyJiEi_ARMIOPQLDXJaf1dU1ZLxyJy3mZk2dJ_fK4r6KKo36OyYXyJShc-D2XHcpDDp1Q7NGayU_uw/s1600/relativistic.png" width="400" /></a></span></div>
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<span style="font-family: inherit; line-height: 115%;">We chose to use the 'proper' measurement units so that everything is nicely symmetrical. Physicists like stuff being symmetrical. The 45-degree
line bisecting the reference frames marks the propagation of a light beam, which should be thought of as a series of absolute events. Note that this line bisects <b>both </b>reference frames.</span><span style="font-family: inherit; line-height: 115%;"> As opposed to the Galilean transformation above, in this <i>relativistic </i>(and <b>correct</b>) transformation, </span><span style="font-family: inherit; line-height: 115%;">the <i>x</i>- and the <i>t</i>-axes of the moving reference frame tilt towards the </span><i style="font-family: inherit; line-height: 115%;">light-line</i><span style="font-family: inherit; line-height: 115%;">, at the same angle. Faster
speeds mean a larger tilt, and in the limit of the speed of Observer 2 going to the speed of
light, the two axes merge:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis1SZdvQTJIZgPWLOZteH0n7p1nhRVTjrj2G102ZmbZz49D9gndaQ9mOiT8PGO125PHpy4Nyps5_zR_mvBzZcVP7P2lf6FwisevBX0hde8dlB7n8C1Ym461PTyqKTJZIyOmqi-X3bM3X0/s1600/fast_faster.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="236" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis1SZdvQTJIZgPWLOZteH0n7p1nhRVTjrj2G102ZmbZz49D9gndaQ9mOiT8PGO125PHpy4Nyps5_zR_mvBzZcVP7P2lf6FwisevBX0hde8dlB7n8C1Ym461PTyqKTJZIyOmqi-X3bM3X0/s1600/fast_faster.png" width="400" /></a></div>
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<span style="font-family: inherit;"><span style="line-height: 115%;">With this in mind, we can see where all the 'weird' stuff in relativity comes from. For example, </span></span><span style="line-height: 18.4px;"><i>simultaneity</i></span><span style="font-family: inherit;"><span style="line-height: 115%;"> can no longer be defined in an absolute (i.e. observer-independent) way.
Events that happen at the same time for one observer can happen at different
times for another one:</span></span><br />
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<span style="font-family: inherit;"><span style="line-height: 115%;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEireNCwQkSJmiBOmDhFaGsCqegdHngYafCyU08S4uS3gT02CD6VLV9hkTP7oCgQKPkI_X7_J0tjwXGfA5Cq89cT0rsh66qcX2FFR5i1LjoU-bnVCUDHofb6VWzBI4qEWkB45yrgilJOzj8/s1600/no_simultaneity.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="262" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEireNCwQkSJmiBOmDhFaGsCqegdHngYafCyU08S4uS3gT02CD6VLV9hkTP7oCgQKPkI_X7_J0tjwXGfA5Cq89cT0rsh66qcX2FFR5i1LjoU-bnVCUDHofb6VWzBI4qEWkB45yrgilJOzj8/s1600/no_simultaneity.png" width="400" /></a></span></span></div>
<span style="font-family: inherit;"><span style="line-height: 115%;"><o:p></o:p></span></span></div>
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<span style="font-family: inherit; line-height: 115%;">Effects like length
contraction and time dilation are very strongly related to the fact that simultaneity (and thus time) is relative. Those can also be inferred by looking at the graphs above and
thinking a bit about how lengths and time intervals translate from one observer to a second, moving one. This is left as
an exercise to the reader. *EVIL LAUGH*</span><br />
<span style="line-height: 18.4px;"><br /></span>
<span style="line-height: 18.4px;">The title of this post comes from a <a href="https://www.youtube.com/watch?v=q2nNzNo_Xps">famous quote by the Doctor</a>, namely, '<i>People assume that time is a strict progression of cause to effect, but actually from a non-linear, non-subjective viewpoint - it's more like a big ball of wibbly-wobbly timey-wimey... stuff.' </i></span><span style="font-family: inherit; line-height: 115%;">One thing that Einsten always
insisted upon, however, was kinda the opposite: the absolute nature of <i>causality</i>. In other
words, he did subscribe to the assumption that the Doctor claims is wrong: the idea that cause always precedes effect. This is what is known as causality, and is usually held as a fundamental principle of nature. An illustration from everyday language is when we say that you cannot eat your cake and have it whole; this is
because the cause - eating the cake - results in the effect - you <b>no longer</b> having
it whole. Einstein made it clear that although some funky things happen
according to STR, nothing as funky as eating a cake and having it whole can <b>ever</b> happen, in any reference frame. Pictorially, this looks like this.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgckJ_G3E2gZgw7E7TulK5uGo8smH4z__lMih_hgnjlIBiPQZTCjjw65RSBpQVpA4tJ1QUcchz5FpWqEI5LUMkG9AaSznep9EOiFRA1h2jn-ul8ZZ7DNu1qE6asCTLYYjxePy2aZmw7NVM/s1600/cake1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="247" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgckJ_G3E2gZgw7E7TulK5uGo8smH4z__lMih_hgnjlIBiPQZTCjjw65RSBpQVpA4tJ1QUcchz5FpWqEI5LUMkG9AaSznep9EOiFRA1h2jn-ul8ZZ7DNu1qE6asCTLYYjxePy2aZmw7NVM/s1600/cake1.png" width="400" /></a></div>
<span style="font-family: inherit; line-height: 115%;">It’s clear that regardless
of the exact velocity of the second observer, the </span><i style="font-family: inherit; line-height: 115%;">world-line</i><span style="font-family: inherit; line-height: 115%;">
of the cake always precedes the </span><i style="font-family: inherit; line-height: 115%;">event </i><span style="font-family: inherit; line-height: 115%;">of
your eating it. Or does it? Remember how the faster the observer moves, the closer their <i>x</i> and <i>t</i> axes come to the light-line, until they practically merge? Well,</span><span style="font-family: inherit; line-height: 115%;"> if we let an
observer move </span><span style="font-family: inherit; line-height: 115%;"><b>faster</b></span><span style="font-family: inherit; line-height: 115%;"> than the
speed of light, and neglect a small </span><i style="font-family: inherit; line-height: 115%;">imaginary</i><span style="font-family: inherit; line-height: 115%;"> detail (we can, cause it’s
imaginary, right?), what happens is what you should expect to happen - the
angle of the tilt of the axes continues to increase, and they switch
places! The space-axis comes above the light line, while the time-axis pops up below. </span><span style="line-height: 18.4px;">The crazy thing is that if you now map the cake, in this new reference frame the event of eating it </span><b style="line-height: 18.4px;">precedes </b><span style="line-height: 18.4px;">the series of events of having it whole:</span><br />
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<span style="line-height: 18.4px;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTQRxgH-hP1f13a0JiP9Vyp3jnh5ySUWGmVYnQIYisKY710zqmtXWyUN-xVHsSkqrqyIg5GCUPNEDzq9GRqD4X4rP7kDmk2kQuv7BlGTe2F9g0U8DoV7uTnIs_68GxqeQDui6qsNgmlKA/s1600/cake2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="235" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTQRxgH-hP1f13a0JiP9Vyp3jnh5ySUWGmVYnQIYisKY710zqmtXWyUN-xVHsSkqrqyIg5GCUPNEDzq9GRqD4X4rP7kDmk2kQuv7BlGTe2F9g0U8DoV7uTnIs_68GxqeQDui6qsNgmlKA/s1600/cake2.png" width="500" /></a></span></div>
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<span style="font-family: inherit; line-height: 115%;">What is perceived as cause becomes an effect, and vice versa: according to this super-luminal observer,
people eat their cakes in order to make them! And then I guess they enjoy them in a moment of un-cooking that </span><b style="font-family: inherit; line-height: 115%;">succeeds </b><span style="font-family: inherit; line-height: 115%;">both eating the cake and having it whole.</span><span style="font-family: inherit; line-height: 115%;"> This sounds like nonsense, and is one of the arguments against the possibility of anything moving faster than the speed of light. But this is a philosophical argument: should we dismiss something just because it <b>sounds </b>like nonsense? There is another strong argument about why we ourselves could never achieve such speeds: the acceleration would require infinite energy, and then some more.
But there is no conclusive argument against the existence of matter that is </span><b style="font-family: inherit; line-height: 115%;">already </b><span style="font-family: inherit; line-height: 115%;">moving faster than the speed of
light. In a similar fashion to us, such matter would not be allowed to </span><b style="font-family: inherit; line-height: 115%;">decelerate</b><span style="font-family: inherit; line-height: 115%;"> to below that speed, as that
would require more than infinite energy. Still, such matter could in principle exist, and, a few decades ago, the theoretical study of <a href="https://en.wikipedia.org/wiki/Tachyon">tachyons</a> - i.e. particles moving
faster than the speed of light - was very hot.</span><br />
<span style="font-family: inherit; line-height: 115%;"><br /></span><span style="font-family: inherit; line-height: 115%;">The enthusiasm has by now somewhat died out, though. Even though we cannot prove them impossible, most physicists don't consider the existence of tachyons very likely. More precisely, the possibility to observe these particles even if they do exist is considered unlikely, which is practically an equivalent statement. That's some food for thought for you: in terms of physics, is there any difference between something not existing, and something existing but not interacting with us?</span><br />
<span style="font-family: inherit; line-height: 115%;"><br /></span>
<span style="font-family: inherit; line-height: 115%;">Einstein would have agreed that tachyons cannot be a part of our world. Basically, interacting with such particles would immediately break causality and result in a ton of paradoxes of the <a href="https://en.wikipedia.org/wiki/Grandfather_paradox">kill-your-grandfather</a> type (and the even more disturbing <a href="https://en.wikipedia.org/wiki/Roswell_That_Ends_Well">Futurama version</a>). Again, this is technically not a proof of the impossibility of backwards-in-time travel. But, while strictly speaking nothing can be proven impossible, some things just look mighty improbable. In any case, Einsten considered the cause-and-effect realtionship a fundamental principle of nature, and its breaking - impossible. In fact he was much more concerned with that than with determinism, despite his famous 'God does not play dice' quote. And I must say that once again I find it easy to agree with the great man. I would say that it is, in fact, safe to </span><i style="line-height: 18.4px;">assume that time is a strict progression of cause to effect</i><span style="line-height: 18.4px;">, at least until we have seen even the tiniest reason to think otherwise. Which, right now, we haven't.</span><br />
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<span style="line-height: 115%;"><span style="font-family: inherit;">Bottom line, while Einstein's theory does illustrate that time is wibbly-wobbly (i.e. not absolute), I don't think he would've been much of a fan of the Doctor's dismissal of causality. </span></span></div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-4128219794453742482015-08-30T09:15:00.003-07:002015-08-30T09:15:59.788-07:00Some stuff Einstein actually said<div class="MsoNormal" style="text-align: justify;">
<span style="font-family: inherit; line-height: 115%;">These blog posts of mine always turn out much longer than I initially expect. What
follows was supposed to be just an introduction, but it became so long that, knowing your attention span (or rather, judging about
it based on my attention span when it comes to blog posts), I thought it best
to just split it off and give it a </span><i style="font-family: inherit; line-height: 115%;">separate entry</i><span style="font-family: inherit; line-height: 115%;"> kind of status.</span></div>
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<span style="font-family: inherit; line-height: 115%;">I’d like to talk a bit more
about space and time (what <b>are </b>those things anyway?), and about absolutes in the theory of relativity. Yes, not
</span><b style="font-family: inherit; line-height: 115%;">everything </b><span style="font-family: inherit; line-height: 115%;">is relative.</span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">By the way, Einstein
himself has written something like a book (it’s more like a collection of lecture
notes) about relativity. It’s called <i>The
Meaning of Relativity</i>. Yes. By Einstein himself. If you’re up to the
challenge, you can try to understand relativity directly through Einstein’s
words. But I wouldn’t recommend that to people who haven’t had at least a basic
course on the subject - otherwise the book will be quite anticlimactic.<o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">Actually, I think the book
is a bit anticlimactic for anyone, since Einstein is such a <b>deity </b>in science that you expect every
word of his to bring you noticeably closer to the true meaning of the Universe.</span><span style="font-family: 'Times New Roman', serif; font-size: 13.5pt;"><o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">In fact the book is a
normal, at times even boring, exposition of relativity, with a good number of
abstruse (also due to their slightly archaic notation) equations. However, he does share some non-mathematical insights, and some of those I just cannot put better than Einstein. Let’s start
with time. He writes,<o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;"><i>The experiences of an
individual appear to us arranged in a series of events; in this series the single
events which we remember appear to be ordered according to the criterion of earlier and later, which cannot be analysed further.</i><o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">While he does not
specifically talk about space (the book is really concise), it is
straightforward to extend the statement in that direction: our experiences
appear to us <b>positionally</b><i> </i>arranged,
so that we can for example say that an object is <i>closer</i> to us than another object. Thus, our concepts for both space
and time stem from our experience. Obviously, these are then always associated
to a particular individual - the one doing the experiencing - and there is <i>a priori</i> no need for any overlap between
various individuals. However, as Einstein writes,<o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;"><i>By the aid of speech different
individuals can, to a certain extent, compare their experiences. In this way it
is shown that certain sense perceptions of different individuals correspond to each
other, while for other sense perceptions no such correspondence can be
established. We are accustomed to regard as real those sense perceptions which
are common to different individuals, and which therefore are, in a measure,
impersonal. The natural sciences, and in particular, the most fundamental of
them, physics, deal with such sense perceptions.</i><o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">This is a truly outstanding
definition of the object of physics. There seems to be some order in the world
in the sense that we <b>all</b> see some things happening in a certain way. Why is that? We don’t really have
to, if you think about it, but the fact that we do implies some underlying laws that unite our
experiences - and that's what physicists try to analyze. </span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">Of course, Einstein allows for experiences which are not shared by various observers, but these are not physical. F</span></span><span style="font-family: inherit; line-height: 115%;">or example, you “totally going all the way with Judy last night” won't be of interest to physicists, especially if it was an experience not even shared by Judy. </span><span style="font-family: inherit;"><span style="line-height: 115%;">Now, an obvious question
springs up. Does this definition of the physical world as 'shared experiences', well, suck, because it's too </span></span><span style="line-height: 18.3999996185303px;">anthropocentric</span><span style="font-family: inherit;"><span style="line-height: 115%;">? And on a related note, is there space and time beyond the human observation of those? Is there
a physical world beyond our perception of it? And how could we ever separate
one from the other? </span></span></div>
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<span style="font-family: inherit;"><span style="line-height: 115%;">What is important to realize is that this question is a </span></span><b style="font-family: inherit; line-height: 115%;">philosophical </b><span style="font-family: inherit; line-height: 115%;">and not a physical one.
And the philosophers <b>have</b> given it a lot of thought - we have Descartes' <i>cogito
ergo sum</i>, which to a first approximation suggests that we can be certain of our
existence (through the very act of questioning it), but of nothing beyond that.
Descartes also has an <a href="https://en.wikipedia.org/wiki/Evil_demon">evil demon</a> that helps him extend that idea. The modern version of the demon is the <a href="https://en.wikipedia.org/wiki/Brain_in_a_vat">brain-in-a-vat</a>, while
the ultra-modern <a href="http://www.imdb.com/title/tt0133093/">Wachowski-siblings version</a> is the backbone to the plot of one of my
favorite movies. Which makes me wonder if The Matrix would've been even cooler if it were called The Evil Demon. </span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">Incidentally,<i> I think therefore I am </i>is a little bit
like the <i>E = mc<sup>2 </sup></i>of philosophy. Many people know it,
but not so many have any idea what the point is. For those, The Matrix is a
great starting point to dive in the implications of the <i>cogito ergo sum</i>. On the
other hand, it’s not a good starting point to learn any physical laws, as they
are consistently broken both in and out of the Matrix... (Still a great
movie!). </span></span><span style="font-family: inherit; line-height: 115%;">Anyway, coming back to physics, I like Einstein’s stance:</span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;"><i>The only justification for
our concepts and system of concepts is that they serve to represent the complex
of our experiences; beyond this they have no legitimacy. I am convinced that the
philosophers have had a harmful effect upon the progress of scientific thinking
in removing certain fundamental concepts from the domain of empiricism, where
they are under our control, to the intangible heights of the a priori. For even
if it should appear that the universe of ideas cannot be deduced from
experience by logical means, but is, in a sense, a creation of the human mind,
without which no science is possible, nevertheless this universe of ideas is
just as little independent of the nature of our experiences as clothes are of
the form of the human body. This is particularly true of our concepts of time
and space, which physicists have been obliged by the facts to bring down from
the Olympus of the a priori in order to adjust them and put them in a
serviceable condition.</i><o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">As usual, I find it really
easy to agree with the great mind. <o:p></o:p></span></span></div>
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<span style="line-height: 115%;"><span style="font-family: inherit;">The main point of this post can then be stated as a summary of everything above. The world in
the framework of the theory of relativity is composed of absolute <i>events</i> which
are embedded in an observer-dependent <i>space-time</i>. The absolute events are the 'shared experiences' that Einstein talks about. Those are observer-<b>in</b>dependent in the sense that everyone agrees that they 'happened'. However, when describing the events, everyone has to inevitably refer to their own space-time reference frame, which is by definition observer-dependent. So, in order for <i>different individuals </i>to <i>compare their experiences</i> <i>by the aid of speech</i>, and find that </span></span><i style="line-height: 18.3999996185303px;">certain sense perceptions of different individuals correspond to each other</i><span style="line-height: 18.3999996185303px;">, a 'common language' is necessary. This provides the means of a translation from one individual's very own, personal space-time to the next. And as I've <a href="http://simplyphy.blogspot.ch/2015/04/a-relatively-special-post.html">mentioned before</a>, </span><span style="line-height: 18.3999996185303px;">the theory of relativity is simply a recipe for how to do the translation.</span></div>
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<span style="line-height: 18.3999996185303px;">Notice then that the big conceptual leap in the theory is </span><b style="line-height: 18.3999996185303px;">not </b><span style="line-height: 18.3999996185303px;">that everything is relative - in fact it's important that there are absolutes, the shared experiences</span><span style="line-height: 18.3999996185303px;">. Also</span><span style="font-family: inherit; line-height: 115%;"> note that we
knew that some things were relative even before Einstein. In particular, we are quite comfortable with the notion that space is relative: if something is two meters away from me, I
wouldn’t generally expect it to be two meters away from you - that would only
be true in some very particular cases. The main innovation of relativity was actually the fact that </span><b style="font-family: inherit; line-height: 115%;">time </b><span style="font-family: inherit; line-height: 115%;">is also relative. We used to think
(and still often do) that if something is two minutes away from my 'now', then
it's also two minutes away from your 'now'. However, just as with space, this turns out
to be true only in some particular cases, which just happen to <b>fully encompass
our everyday experiences</b>, which is why it took us so long to figure that out. Better said, it’s not possible to define our 'nows' in a way that they match under all circumstances (in particular when we move
fast with respect to one another). Better still, space and time are inseparable
and kinda mixed up and only kinda absolute if thought of as a single unit: space-time.
However, they are always relative when separated. </span></div>
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<span style="font-family: inherit; line-height: 115%;"><br />OK, this is getting a bit too technical. But I’ll try to illustrate all these
points even further in the next post.</span></div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-44780675216692222242015-05-30T23:56:00.000-07:002015-05-30T23:56:46.332-07:00'Energy' is a great name for a band*<div style="text-align: justify;">
<span style="font-family: inherit;"><sup>*</sup>Actually not really. It sounds cool but it would be impossible to look up on google, and anyway I would guess it's already being used. But I don't know cause it's hard to look up on google. </span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Where was I? Ah, yes: there's a fascinating
relationship between the microchips that allow you to read these words and the
colorful patterns on the wings of a butterfly.</span><br />
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<span lang="EN-US"><span style="font-family: inherit;">The original idea of this post
was to explain what I'm working on, at least qualitatively. I thought a good introduction
to that could be the wings of a butterfly, but then I realized an even better
starting point would be the unlikely relationship that they share with
microprocessors, so I thought I'd start off there... and then I finished the
post and never got to talking about what I'm working on, and neither did I get to the
butterfly. But I will get to both, soon. <o:p></o:p></span></span></div>
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<span lang="EN-US"><span style="font-family: inherit;">Anyhow, hopefully I got you interested. Good. Now I'll go into
the boring stuff. Semiconductors seem to be boring. Honestly, even I was bored
when I first had to learn the details. Think about it - do you even have an
idea what a semiconductor is? My observations are that the majority of
scientifically interested people (not physicists, though), who have an idea
about quantum mechanics and relativity and superconductivity and dark matter
and the Higgs boson, have a hard time defining what a semiconductor is, and,
consequently, why they are the backbone of practically any gadget around
us. Occasionally someone would know that
a semiconductor is a material with conductivity between that of a conductor and
that of an insulator, but a fairly intelligent person can practically read that
off their name. Essentially, they're not quantum-black-hole-God-particle-wormhole
material, they’re boring stuff... which has meanwhile had more impact on our lives than literally anything else in literally the
whole history of humanity. So I'll dare
try to lay out the basics here - please bear with me if you start yawning.<o:p></o:p></span></span><br />
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<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Let's start </span><b style="font-family: inherit;">all </b><span style="font-family: inherit;">the way back: throwing a ball. Or a
potato. When you throw one (or the other), there is a fixed relationship between its energy
and its momentum. The harder you throw it, both of those increase, but the
energy increases with the velocity </span><i style="font-family: inherit;">v </i><span style="font-family: inherit;">squared
(</span><i style="font-family: inherit;">E = mv<span style="mso-text-raise: 4.5pt; position: relative; top: -4.5pt;">2</span>/2</i><span style="font-family: inherit;">), while the momentum scales
just with </span><i style="font-family: inherit;">v </i><span style="font-family: inherit;">(</span><i style="font-family: inherit;">p = mv</i><span style="font-family: inherit;">). So, if we were to draw a graph
of energy versus momentum, it would look something like that. </span></div>
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<span lang="EN-US"><span style="font-family: inherit;">This holds true for any
'classical' (= large enough) moving object, but it turns out that even
with quantum mechanics taken into account, the energy-momentum graph for
massive elementary particles still looks like the one above! If you're ever asked
to name one similarity between an electron and a potato, you're welcome. No,
really, if you do get that question and say that energy is proportional to the
momentum squared, you'll make an impression. <o:p></o:p></span></span></div>
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<span lang="EN-US"><span style="font-family: inherit;">By the way in line with my
previous posts, do notice that this does not pre-suppose any particle nature of
the electron - energy and momentum can be defined independently of that. However,
in both cases the graph above only holds true when we consider free propagation.
If<i> </i>there is interaction with the
surrounding world, the graph could change significantly. In most situations of
practical interest, this is straightforward to take into account for potatoes
and other objects from the classical lot, but elementary particles become much
more tricky when interactions are turned on. To make matters worse, electrons
love to interact with their environment, because they are charged particles,
and there are usually other charged particles around them (including, but not
limited to, other electrons). This makes characterizing the way electrons move
in a given material an extremely difficult task. On the other hand, it is
arguably the most important characterization we could do, since a large number
of a material’s physical, chemical, optical, and electrical properties can at
least in principle be extracted on this basis. And so, this study, usually
referred to as solid-state physics (the term is a bit broader than just the
motion of electrons, but that's the gist of it), has produced an astonishing
number of PhD theses. <o:p></o:p></span></span></div>
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<span lang="EN-US"><span style="font-family: inherit;">Now, in all crystals, which is
to say materials with some structure in them, you have a lattice of ions
through which the electrons propagate. This of course modifies the
Energy-momentum graph, and it starts looking somewhat like this. <o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjk5Es3I9Jf-_lgGhHQ3311dg4dFelAKhA2wrc8eBcR9lubuK4aP4wYtbSavDwLxcmadWtPID11eeWKNzxgLsJLDAAu40eyHCmL0aNu_kg4pkLgCxvWegUsalkT4ac98XJ3zsfZpQRyO1Y/s1600/bands_unfolded.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="230" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjk5Es3I9Jf-_lgGhHQ3311dg4dFelAKhA2wrc8eBcR9lubuK4aP4wYtbSavDwLxcmadWtPID11eeWKNzxgLsJLDAAu40eyHCmL0aNu_kg4pkLgCxvWegUsalkT4ac98XJ3zsfZpQRyO1Y/s400/bands_unfolded.png" width="400" /></a></div>
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<span lang="EN-US"><span style="font-family: inherit;">The jump discontinuities occur at exactly the same intervals, and so the graph above is usually drawn in the
"folded" version, like this<o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjr0sZi4GKfLwcN3WvL6l1KVm3aqPvglTc41Jtbwxf6fyrnx9ph7fKH527dEhaVcKrNGM2ijKCq0Xujc5bf3D8U2BjQMjpQNqRQ8KMpwC98-XD0ZVTQC342GEtOh0MXJRpQ1y4NZop98Nw/s1600/bands_folded.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="249" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjr0sZi4GKfLwcN3WvL6l1KVm3aqPvglTc41Jtbwxf6fyrnx9ph7fKH527dEhaVcKrNGM2ijKCq0Xujc5bf3D8U2BjQMjpQNqRQ8KMpwC98-XD0ZVTQC342GEtOh0MXJRpQ1y4NZop98Nw/s320/bands_folded.png" width="200" /></a></div>
<span style="font-family: inherit;">It is easy (well, for a
physicist, at least) to know where to 'fold' the graph - the fact
that the crystal lattice is periodic ensures the periodicity of the 'special points' on the momentum axis, and you fold at the first one of those. This often has some deep physical
significance, but anyway, you could as well just think of it as a more compact
representation of the full plot.</span></div>
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<span lang="EN-US"><span style="font-family: inherit;">Hope you're still with me here
– we're getting to the point where you'll understand the title of the post. The
energy intervals within which an energy-momentum graph exists are called 'energy bands', cause, well, they kinda do look like bands when you highlight them:</span></span><br />
<span lang="EN-US"><span style="font-family: inherit;"><br /></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxDIlklbqy2tILbdOELhNSCjbWOMJ6_Xrwvs2ZSr9RSfTsdLyqnrv72QE_WyoM2sintiJIusjFUd7rIYV0GP8Pbre2Z_7fzJ5d6c63ZlLKpyDfQrBvyRDC4kbaaX39PKCkuzVrZEIClPo/s1600/bands.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="221" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxDIlklbqy2tILbdOELhNSCjbWOMJ6_Xrwvs2ZSr9RSfTsdLyqnrv72QE_WyoM2sintiJIusjFUd7rIYV0GP8Pbre2Z_7fzJ5d6c63ZlLKpyDfQrBvyRDC4kbaaX39PKCkuzVrZEIClPo/s320/bands.png" width="320" /></a></div>
<span style="font-family: inherit;">How much of the energy spectrum is covered by these bands, and in what way, determines a lot of the properties of the material. In fact, perhaps even more
importantly – although technically carrying the same information – are the
regions outside the energy bands, which are called band gaps:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7TtW3ErozX9ialceC-VdfEpGFPLZGvMU2cabyuSdK7xJfyHVDraOluixTG7PGLCVC8Ayu-oRKh9EYteEmKl-GB4w-JCzUkewA9DkwiA2NPk7-sAe12VrMFr7CALwx7-Tp5Q4TFnGJ_VI/s1600/band_gaps.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7TtW3ErozX9ialceC-VdfEpGFPLZGvMU2cabyuSdK7xJfyHVDraOluixTG7PGLCVC8Ayu-oRKh9EYteEmKl-GB4w-JCzUkewA9DkwiA2NPk7-sAe12VrMFr7CALwx7-Tp5Q4TFnGJ_VI/s320/band_gaps.png" width="320" /></a></div>
<span style="font-family: inherit;">The significance of those is a
fairly bizarre one: electrons of that energy simply cannot exist in the
material! That's now quite different from throwing a potato – you can throw it
with any speed you like, and it will propagate with the corresponding kinetic
energy – there are no forbidden energies. That's obviously not always the case for an electron
in a crystal. Now, remember that electrica current is nothing else but motion of electrons. Because of this, the energy bands - and the presence of band gaps - fully determine whether a material is a
conductor, a semiconductor, or an isolator. To see this, we need to add the
last piece to the story, which is the fact that in every material there are a
number of electrons that could move around, if 'pushed' (which in physical terms is done by applying voltage). If they're not pushed, they occupy all the
lowest energy 'states' – this is because nature in general is quite lazy, and
does everything with the lowest possible amount of energy, unless it has a very good reason to do otherwise. Now, every material can be characterized by a certain
number called the Fermi energy, whose significance is that there are electrons
with all energies below that level (and none above, at least not at 0 temperature). This together with the 'band
structure' of the material determine its electrical properties in the following way. </span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrw3fVplKE_WRKsyIGsAb0FTWM8CrsjPk9TDkFefXGB11DV_XMvpnVaD1BNlMGp7wPzPwn_FMnYFVMcSFI1q3p-slusOtW94ThnP8q02mQomeZF7R86aVtpUgwC3OYp90FVe1qZJrh53c/s1600/fermi.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="201" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrw3fVplKE_WRKsyIGsAb0FTWM8CrsjPk9TDkFefXGB11DV_XMvpnVaD1BNlMGp7wPzPwn_FMnYFVMcSFI1q3p-slusOtW94ThnP8q02mQomeZF7R86aVtpUgwC3OYp90FVe1qZJrh53c/s400/fermi.png" width="400" /></a></div>
In brief, if the Fermi level crosses a band, then arbitrarily small voltage makes the electrons move, i.e. creates current, which the material 'conducts'. If the Fermi level is in a band gap, the applied voltage needs to be large enough so that the difference in energy between the Fermi level and the closest higher-energy band can be overcome. Thus, t<span style="font-family: inherit;">echnically, there is no
difference between a semiconductor and an isolator: it is in principle possible for both of them
to conduct electricity, but only if enough voltage is applied. What makes a material an insulator is then only the practical
consideration that an impossibly high amount of voltage is needed to make its
electrons move in a certain direction, while for a semiconductor that value is achievable. </span></div>
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<span lang="EN-US"><span style="font-family: inherit;">Now, another important aspect
of semiconductors which ultimately renders them useful for CPUs is that they
can be doped<o:p></o:p></span></span></div>
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<span style="font-family: inherit;">The red dots in the image above represent 'impurities' which are artificially introduced to make the semiconductor either more or less conductive (we can do both). I'll skip the rest of the details: the important point is that by making contacts ('junctions') between differently doped semiconductor pieces, you have a lot of non-trivial control on the way current flows through a device. Two examples are diodes (the D in LED) and transistors. The latter, very schematically, work in the following way</span></div>
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<span lang="EN-US"><span style="font-family: inherit;">Essentially, the transistor is a 'switch': if there is no current at the 'gate', the switch is off, and no current can pass from the source to the drain. The switch can be turned on by current flowing at the gate. And that's all you need to make
a logic circuit! No current is your 0, yes current is your 1. You connect the
drain of one transistor to the gate of another, and your 0 or 1 influences the
next 0 or 1. Depending on how you make the connections, there's no end to the
possibilities of the allowed operations that you can hard-wire in a chip. Of
course you only go for some basic ones, and then leave it to a programer to
play around in arranging those in ever-more-complicated patterns, so that you
can all enjoy your cat pictures and your instant connectivity with anyone
anywhere around the world and your daily dose of videos of naked people doing
dirty stuff and reading this post and God knows what else you're into that's
only made possible because of what I explained here.<o:p></o:p></span></span></div>
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<span lang="EN-US"><span style="font-family: inherit;">Hope it wasn't so boring after
all.</span></span></div>
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Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-19213472272926071332015-05-10T10:23:00.000-07:002015-05-10T10:23:10.797-07:00...and glory to the quantum! <div style="text-align: justify;">
(<i>part 2</i>)</div>
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If you’re still not convinced that the <a href="http://simplyphy.blogspot.com/2015/04/down-with-duality.html">wave-moose</a>, I mean, the wave-particle duality is confusing and not even well-defined, consider this: there's not even a consensus on how much of each part makes up the quantum entity (a bit like <a href="http://en.wikipedia.org/wiki/Person_of_Christ">Jesus</a>). According to Bohr, it is strictly a wave <i>or</i> a particle, never both. According to de Broglie it is both (a particle guided by a wave), and according to other people (like Penrose), it is neither (the duality principle is just an illustration, not reality). And intermediate positions have also been expressed by pioneering physicists. Here, I’m going to expand on the neither-nor viewpoint, which so far only says what the object is<b> </b><i>not</i>, and not what it <i>is</i>.</div>
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'It' is a quantum. The word 'quantum' is now much more commonly used as an adjective than as a noun. However, in its <a href="http://en.wikipedia.org/wiki/Quantum#Etymology_and_discovery">initial meaning</a> it <i>is</i> a noun, and when we describe how this noun behaves, we are describing its… mechanics! This is the way in which I understand the term Quantum Mechanics, and it's my conjecture that this is how it was (etymologically) meant to be understood. In fact, in old papers I have seen authors talking about 'the quantum' in the same way that we nowadays talk about 'the particle'. Now, of course words have no intrinsic meaning hard-wired into them - it is us who load them with meaning. In that sense, there is no a priori reason why a 'quantum' is a better term than a 'particle' or a 'qauntum particle', but there is a pretty good a posteriori reason: the word 'particle' is already loaded with too much meaning - it inevitably evokes a picture of something round, solid, and localized in a tiny region of space, and none of these properties are necessarily properties of the quantum. </div>
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So what is the quantum? It's hard to define it with no mathematics and in just one sentence, but let's say that it's <i>the smallest amount of energy that can exist on its own</i>. However, there are different types of <i>quanta</i> – an electron quantum, a photon quantum, etc., which we commonly refer to as different particles. A crucial difference between the former and the latter is that the quantum does not need to be localized at a particular point of space. It could be, but it doesn’t have to be. In fact, its natural state is not localized, but due to the fact that quanta interact with each other, it's hard to isolate them into this natural state. Thus, in reality - and in experiments - a quantum often appears localized, but only because there is an external force that confines it. Now, it is often said - by people I do respect as scientists, mind you - that the de-localized nature of quantum objects is completely non-intuitive: <a href="https://www.youtube.com/watch?v=9RExQFZzHXQ">here</a> are just two examples <a href="https://www.youtube.com/watch?v=JKGZDhQoR9E">where</a> I recently heard that statement. The argument is that it is non-intuitive because we never experience anything like it in our lives. But this is a poor argument: the fact that the Earth is round and rotating, that a bowling ball and a feather in vacuum fall in the exact same way, or that a body in motion will stay in motion unless an external force stops it, those are all aspects of nature that we never experience directly but can only infer from observations. Yet we never say that they are non-intuitive, in fact we never even question them and have accepted them as almost mundane.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7EusqtKu2vcG8yd4lANNkwvSSqs0YfO-hyrXRHyVSlNDaqw0j-c7tW1ga_Pz8SSob7LtbIku_sp0ToYNo5iK02XIXToUcEFviP1oSDRqZ82p9hEk7WmVPU82JUuXm0yRtqRpT7gKRkYE/s1600/stoner1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="171" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7EusqtKu2vcG8yd4lANNkwvSSqs0YfO-hyrXRHyVSlNDaqw0j-c7tW1ga_Pz8SSob7LtbIku_sp0ToYNo5iK02XIXToUcEFviP1oSDRqZ82p9hEk7WmVPU82JUuXm0yRtqRpT7gKRkYE/s400/stoner1.png" title="Just play Shine on You Crazy Diamond and you will, too" width="400" /></a></div>
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I see a sort of a vicious circle around Quantum Mechanics: by now it's practically a cliche to say that it is non-intuitive, and this is repeated and perpetuated every time the theory is mentioned. However, there is nothing intrinsically non-intuitive in the theory, because intuition - just like the meaning of words - can evolve, and has evolved many times in the history of science. However, a prerequisite for QM to ever become intuitive is that we stop reiterating that it's not. </div>
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In fact, I think there is a curious analogy to be found in the history of science. Insisting that objects are naturally localized into 'particles' is very similar to insisting that an object's natural state is at rest. There are forces acting all around us, so it looks as if everything eventually comes to a rest, if there's nothing pushing it. But if you remove the surrounding forces (as Newton realized and postulated), an object will continue moving indefinitely. Everyone accepts this today, although it would have sounded very non-intuitive to Aristotle and everyone else in his generation. In the same sense, we are used to objects being localized, because electrons interact with nuclei to form atoms that interact with other atoms to form molecules that interact to form, well, everything around us. What Quantum Mechanics simply tells us is, the localized nature of everything around us is not its natural state; it is due to interactions. Remove interactions, and the 'particle' spreads out and is no longer a 'particle' in the sense we would typically attach to that word.<br />
<br />
We actually <i>do</i> observe this all the time: light is made out of quanta that interact weakly with other quanta - in most situations, in fact, negligibly. This is why we are used to thinking of light as waves - it is much closer to its natural (by which I mean non-interacting), de-localized state, since there are no forces confining it. When we do the double-slit experiment that I outlined in part 1 of this post, the photon quanta first propagate freely and look like waves, but then they interact strongly with the detector in the end - e.g. a photographic film in which they get absorbed - making them appear localized. This property looks particle-like, but really we always have the same object - a qauntum - but in the presence or in the absence of interactions. Eventually, the jump in intuition required to accept that fact is no bigger than the one that was needed to accept Newton's principle that an object will keep on moving if there's no interaction to stop it. </div>
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The current intuition that stuff should ultimately be localized comes, of course, from trying to imagine everything as particles. Why this obsession? Why think of anything as a particle? Why do we say it's hard to imagine a baseball as a wave? Why don't we just imagine everything as a wave, or better: as energy! That's what Einstein tells us anyway! (E = mc<sup>2</sup>)The Earth is not a hard sphere! Neither are atoms nor nuclei nor anything. They are all a bunch of energy clustered together because it attracts itself. How do you define 'hard' anyway? How would you define the size of a particle if you want to stick to that notion? You can never actually touch it, you can only get to a given distance before the repulsion gets too strong. The Earth appears hard because, while it attracts us on the large scale, it's repulsive on the short scale (due to electrostatic repulsion between our atoms and Earth’s atoms). Most of us know all of those individual facts, yet strangely cling to the image of everything as made out of tiny matter-balls that we call particles.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaT5014hgmMATkDNgeEGf7CLxj7u9xiuH0MAMBn67oJr35I6d6rZ2GEvmKqn1A2YG5cFd5A2DDipKCQ_cBGgRAkBnUuk3Cv4npcKKWkvZybaUY-INb4-7JEGZgOyVplxfwxmAOy76ZxvU/s1600/morpheus.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="156" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaT5014hgmMATkDNgeEGf7CLxj7u9xiuH0MAMBn67oJr35I6d6rZ2GEvmKqn1A2YG5cFd5A2DDipKCQ_cBGgRAkBnUuk3Cv4npcKKWkvZybaUY-INb4-7JEGZgOyVplxfwxmAOy76ZxvU/s400/morpheus.png" title="Real life is even cooler than the Matrix, and that's saying something!" width="400" /></a></div>
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There's nothing that's actually solid. Matter is energy (is quanta of energy). If anything appears solid to you, it's because the energy that constitutes it is repulsive to the energy that constitutes your hand.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJawoqJv_nvMjiAkjLY5TVrowOZmpkxwprrrbjgu4Xj5Wv6t8LHKOlBE1HUnJgE8OPiHgmRpoYkspjw2UaTR_Ox-Au2sn2s9KJG6EK3Yir7iT_gxYf7Y8ov8rdW5Qh3PL65d5lYcCp2FI/s1600/stoner2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" height="151" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJawoqJv_nvMjiAkjLY5TVrowOZmpkxwprrrbjgu4Xj5Wv6t8LHKOlBE1HUnJgE8OPiHgmRpoYkspjw2UaTR_Ox-Au2sn2s9KJG6EK3Yir7iT_gxYf7Y8ov8rdW5Qh3PL65d5lYcCp2FI/s200/stoner2.png" title="Yep. " width="200" /></a></div>
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Maybe right now this sounds hard to grasp, and you're thinking -'what's the use, if it's abstruse?' (no rhyme intended). Could one argue that better intuition is gained in the wave-particle picture than in this 'quantum' picture? My answer to that is another question: why is then 'non-intuitive' the most common adjective assigned to Quantum Mechanics? My argument is that this is largely because of trying to describe an object by starting from the notion of a classical particle, and then outlining all the aspects in which it's <i>not</i> like one. Isn't it better to just define the object through its properties, complicated as they might be? Isn't that the way to break the vicious 'non-intuitive' circle?</div>
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I want to discuss in more detail said properties of the quantum, and the way it compares to the wave-particle view, as well as the difference between the quantum and the wave function. But in the interest of me being terribly late with new posts, I'll leave this for a future part 3. </div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-29655206789930931882015-04-19T06:06:00.000-07:002015-04-19T06:06:30.607-07:00Down with duality...<div style="text-align: justify;">
(<i>part 1</i>)<br />
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Where I come from, we make jokes, for some absolutely unknown to me reason, with people from the <a href="http://en.wikipedia.org/wiki/Chukchi_Peninsula">Chukchi Peninsula</a> located at 'the northeastern extremity of Asia.' One example is the following joke. A guy from the region goes on a trip to Africa, and when he comes back, the whole village gathers to hear his stories. Says he, 'I saw a giraffe!' but the people don't know what that is, so they ask. He replies, 'well, you know what a moose is, right? A giraffe is like a moose with a really long neck.' Everyone is amazed and asks what else he saw. 'I saw a zebra' - adding, due to the bewildered faces around - 'it's like a moose, but black and white.' They want to hear more, so he says, 'I also saw a crocodile.' He stops, thinks for a while how to describe this one, and says, 'well, you know what a moose is. Imagine something that has nothing in common!'</div>
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I think something similar commonly happens when we speak of quantum mechanical effects, especially when the wave-particle duality - the notion that quantum mechanical objects sometimes behave as waves and other times as particles - is invoked. The idea that stuff around us is ultimately made out of 'particles', in the sense of tiny balls of matter, is heavily ingrained in our worldview. For example, even though we know that the picture of an atom with the electrons orbiting as planets around the nucleus is wrong, it's what everyone imagines when thinking about atoms, and especially Big Bang Theory fans who see this at every cut-scene. There is this general peculiarity in the thinking about all small-scale physics, even when the thinking is done by physicists: we know that it is the wave function that rules the world on those scales, but we try to translate that world into something that's made out of particles, which just happen to have some weird properties. This inevitably and quickly results in statements of the form, 'a quanum mechanical particle is nothing like an ordinary particle in that...' We end up describing something not by defining its characteristics, but by outlining the characteristics by which it <b>differs</b> from something that we are very familiar with... just like the Chukchi peninsula guy in the joke. I find this equally absurd.<br />
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This is again due to the Copenhagen interpretation, and came about in the following way: Niels Bohr insisted on separating reality from the wave function; for him, reality was made out of observable objects, and the wave function was simply a tool to (probabilistically) predict how those behaved. It was important, however, to always discuss quantum physics in classically accessible terms, which is why he introduced the concept of <a href="http://en.wikipedia.org/wiki/Complementarity_(physics)#Concept">complementarity</a>, one manifestation of which is the wave-particle duality. In short, Bohr's view was, never mind the mathematics; that's not reality. Reality is anything that would come out in a measurement, and, in terms of that, the same quantum mechanical object sometimes has properties of a wave and sometimes has properties of a particle (in the sense that we are familiar with from classical physics).<br />
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My claim is that this has <b>hindered </b>the understanding of QM, rather than helped it, and that we need a revolution.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLHfuf2AAJ97PKuew56o1lqhtXd8osXcbK5eI3dOOpwP8s8xpk94gGj_QBsBeDzu3J1BIdQSeUd9WbTPENx5PzgjaJIzoPdnl2dq7G2W-YGsMQKK7vvZw9dtUqBzukkh-bRjYyxrnY1kI/s1600/revolution.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLHfuf2AAJ97PKuew56o1lqhtXd8osXcbK5eI3dOOpwP8s8xpk94gGj_QBsBeDzu3J1BIdQSeUd9WbTPENx5PzgjaJIzoPdnl2dq7G2W-YGsMQKK7vvZw9dtUqBzukkh-bRjYyxrnY1kI/s1600/revolution.png" height="277.5" title="I know! We should've gone to that "Schroedinger's cat is not dead and alive" event instead!" width="500" /></a></div>
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A good illustration comes from an experimental result which recently went viral in science-oriented social media. I will use <a href="http://phys.org/news/2015-03-particle.html">this</a> discussion of it to deconstruct the misconceptions, but I want to highlight that I am not trying to criticize neither the particular website nor the particular research group; the problem is much more fundamental, and this is just a handy, recent example. I encourage you to have a look at the article before continuing reading. It's a very sexy piece of news as far as developments in Physics are concerned, hence I'm not surprised by the attention it received. But, essentially, I have a problem with each of the first three sentences of the news article.<br />
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'<i>Light behaves both as a particle and as a wave.</i>' This is only within the Copenhagen interpretation, and in addition only within its very classical formulation. It is certainly what we all get taught in Physics classes, but anyone who has made one step further in thinking (or reading) about the philosophy of quantum mechanics might have questioned the statement. But, more importantly, the wave and particle behavior is essentially how some people choose to visualize quantum mechanical objects - it is <b>not </b>a scientific truth in any sense because it isn't even <b>rigorously </b>defined. There is no mathematical formulation of the principle; instead, it is used in 'hand-waving' explanations with the goal of providing some intuition. Occasionally, this works quite well, and I guess it was in particular helpful in the early days of the theory. I've just started feeling like we've outgrown it.<br />
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'<i>Since the days of Einstein, scientists have been trying to directly observe both of these aspects of light at the same time</i>.' They sure haven't. Or, <span style="color: blue;">[citation needed]</span>. Physicists generally realize that the wave-particle duality is meant to help us visualize the behavior predicted by the wave function, but I had never heard anyone discussing capturing both aspects at the same time as an important experimental goal. Actually that 'goal' is, again, not even <b>strictly defined</b>. Which brings me to the next point.<br />
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'<i>Now, scientists at EPFL have succeeded in capturing the first-ever snapshot of this dual behavior.</i>' First of all, what does a '<i>snapshot</i>' even mean, in this case? You probbly know that in order to take a photo, you always need some exposure time - the same is true here. I don't want to explain the details of the experiment - you can read the article and watch the video, I think it's fairly accessible. The bottom line is, however, that they had to average over many electrons interacting with many photons to capture the 'snapshot', so, even though everything is happening fast, is it really 'snap'?<br />
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Going with this train of thought, I have two questions to that statement: have they? And is it the first-ever? The answers cannot be both 'yes'. If this experiment counts as a snapshot of light as both a wave and a particle, then so should the following one, which is so old, fundamental, and pioneering, that it is by now a <a href="http://teachspin.com/instruments/two_slit/index.shtml">classroom experiment</a>. I'm talking about the famous double-slit, which goes like this: you shine light towards a screen with two holes in it, and record the intensity at a certain distance behind.<br />
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It has been known already for centuries that the intensity presents an 'interference pattern' of alternating minima and maxima, since the waves coming from each slit add up in either a constructive or a destructive fashion. Now, if you decrease the intensity <b>a lot</b>, at some point what you start recording begins to look like discrete points on your screen - we would call those photons, or 'quanta' of light. The funny thing is that if you record many of them and average out, you would still get the same interference pattern, even though you were recording them one by one, as 'particles'.<br />
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The same experiment can be done with electrons, in which case it is supposedly more striking, because, supposedly, we are used to thinking of light as a wave and of electrons as particles. But, essentially, they are both quantum mechanical objects, so no wonder they behave in the same way! Anyway, the same thing happens also with electrons, and you can see a nice <a href="http://en.wikipedia.org/wiki/Double-slit_experiment#/media/File:Double-slit_experiment_results_Tanamura_2.jpg">experimental result</a> of the build-up of the interference pattern. So, doesn't this experiment also produce a 'snapshot of the dual behaviour?' I see no reason not to count it as one.<br />
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In fact, in the double slit, it is quite clear what the wave and what the particle aspects are. But have a look at the 'snapshot' from the recent experiment that I discussed above:<br />
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I think an obvious thought is, 'I see the wave, but where is the particle?' That sort of confusion can for example be seen in most of the <a href="http://www.reddit.com/r/science/comments/2xq5nf/the_first_ever_photograph_of_light_as_a_particle/">reddit comments</a> on the article. It's natural to not see the 'particle' if you're not a specialist, and it's difficult even for specialists, because the axes are not labelled in this photo. One of the axes represents position, but the other actually represents energy. What is demonstrated is, in fact, a wave-like interference pattern together with <i>quantization of energy</i>, something I explained <a href="http://simplyphy.blogspot.ch/2015/02/ladder-to-heaven.html">here</a> (the photo above looks quite similar to the guitar string vibrations, doesn't it?). In fact, if you look at the title of the actual <a href="http://www.nature.com/ncomms/2015/150302/ncomms7407/full/ncomms7407.html">research article</a> (as opposed to titles in popular-science media), it says <i>quantization</i>, not <i>particle</i>. This is because the former is well-defined mathematically, while the latter is not.<br />
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The two experiments above do serve as a good illustration of the two properties that are most commonly associated with 'particles'. One, a particle is something which is well-localized in some finite region of space (double slit experiment), and two, it is something that comes in integer numbers - you can have one or two or three of 'em, but not one and a half (both experiments). But - think about it - both of those statements hold true for a moose as well! Is it an equally valid viewpoint, then, to imagine reality made out of tiny meese<sup>*</sup>, and talk about a wave-moose duality? I leave it to you to decide.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7E4kQ2vWY5tS5nReuKk4z9SEPDnVpPua_lCdb82Kbwc633EoAXXTlP3KzvLaJ1dLr_oYhEsp3aqB0v4fYpafNDR6gZllub6TIo3wOINcAdr4QEiXWckSC9IPtwYgtkFxCWK2xj2er1CY/s1600/wave-moose.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7E4kQ2vWY5tS5nReuKk4z9SEPDnVpPua_lCdb82Kbwc633EoAXXTlP3KzvLaJ1dLr_oYhEsp3aqB0v4fYpafNDR6gZllub6TIo3wOINcAdr4QEiXWckSC9IPtwYgtkFxCWK2xj2er1CY/s1600/wave-moose.png" height="217" title="If we do take action, the mighty wave-moose might be extinct by 2025! Let's do it!" width="320" /></a></div>
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In short, the wave-particle duality is not rigorously defined, leads to a lot of confusion, and would be equally valid if it we replaced it with a wave-moose picture... Given all that, I say, to hell with duality! We must define objects by their properties, not by the properties by which they differ from objects that we are used to! There <b>are </b>alternatives. Several, in fact! If you are a fan of waves, the Everett interpretation is the way to go. If you really want to keep imagining particles, Bohmian mechanics would be your thing. I will explain both of those in due time, but first I'd like to finish with the basics of QM within the standard interpretation. In the second part of this post, I will thus give an alternative, which, like freetown <a href="http://en.wikipedia.org/wiki/Freetown_Christiania">Christiania</a>, stays within the boundaries of Copenhagen, but goes against some of the accepted norms.<br />
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* <i>It seems that 'moose' is the most standard plural of the word, but <a href="http://en.wiktionary.org/wiki/moose">wiktionary</a> says that, 'the form meese [...] will in most cases be greeted with a snicker, and is thus generally only appropriate in humorous contexts,' which suited perfectly my intentions. </i></div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-54676456542664779462015-04-10T01:54:00.000-07:002016-02-14T03:14:17.947-08:00Does this post shave itself? <div style="text-align: justify;">
In case you're wondering, the title alludes to <a href="http://en.wikipedia.org/wiki/Russell%27s_paradox">Russell's paradox</a>.</div>
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A catalog of all posts (newest first) follows. </div>
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<b>Quantum Mechanics</b><br />
10.05.2015 : <a href="http://simplyphy.blogspot.ch/2015/05/and-glory-to-quantum_10.html">...and glory to the quantum!</a> : The mind-blowing second part of my rant against wave-particle duality.<br />
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19.04.2015 : <a href="http://simplyphy.blogspot.ch/2015/04/down-with-duality.html">Down with duality...</a> : Wave-particle duality leads to a lot of confusion, and is not even something that's strictly defined! Do we really need it? Or has our understanding come to a point where we can do better?<br />
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08.03.2015: <a href="http://simplyphy.blogspot.ch/2015/03/jigsaws-cat.html">Jigsaw's cat</a> : A discussion of the famous feline thought experiment and how it uncovers the measurement problem of the Copenhagen Interpretation (which is easily the main problem).<br />
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01.02.2015: <a href="http://simplyphy.blogspot.ch/2015/02/ladder-to-heaven.html">Ladder to heaven</a> : A description of what is meant by 'quantization', and how it arises from one very special property of the wave function. Visualized through a quantum guitar. </div>
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30.01.2015: <a href="http://simplyphy.blogspot.ch/2015/01/a-trip-to-copenhagen.html">A trip to Copenhagen</a> : A statement of the postulates of the Copenhagen Interpretation. An auxiliary (= boring) post with some necessary technicalities... and two jokes in the end. </div>
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10.01.2015: <a href="http://simplyphy.blogspot.ch/2015/01/to-be-or-to-not-be.html">To be or to not be</a> : Outlining some common misconceptions (and pet peeves!) arising from asking questions which are meaningless in the framework of the Copenhagen (= standard) Quantum Mechanics. </div>
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<b>Relativity</b><br />
14.02.2016: <a href="http://simplyphy.blogspot.ch/2016/02/gravity-waved-q.html">Gravity waved! (Q&A)</a> : The most recent big result in Physics (at the moment of writing of this sentence) is the gravitational wave detection by the LIGO experiment. Tune in to this Q&A to learn more!<br />
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21.09.2015: <a href="http://simplyphy.blogspot.ch/2015/09/wibbly-wobbly-timey-wimey-stuff.html">Wibbly-wobbly timey-wimey... stuff</a> : Some graphical illustrations of space and time in relativity. Time is indeed wibbly-wobbly, but still a progression of cause-and-effect, it seems.<br />
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30.08.2015: <a href="http://simplyphy.blogspot.ch/2015/08/some-stuff-einstein-actually-said.html">Some stuff Einstein actually said</a> : A discussion of space and time, which are both relative, and events which we all agree happened, which are absolute. Motivated by Einstein's book <i>The Meaning of Relativity</i>.<br />
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02.04.2015: <a href="http://simplyphy.blogspot.ch/2015/04/a-relatively-special-post.html">A relatively special post</a> : Special relativity 101 - how it came about, what motivated Einstein, and what the theory is about, in a nutshell. </div>
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<b>Miscellaneous</b><br />
18.01.2016: <a href="http://simplyphy.blogspot.com/2016/01/let-me-teal-you-about-cyansce-of-color.html">Let me teal you about the cyansce of color</a> : What are the intrinsic characteristics that give objects their color? What color is a mirror, and why is the sky blue (sometimes)?<br />
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31.05.2015: <a href="http://simplyphy.blogspot.ch/2015/05/energy-is-great-name-for-band.html">'Energy' is a great name for a band*</a> : What exactly is a semiconductor, is it really different from an insulator, and why in the world has it had more influence on our lives than anything in the history of humanity, ever?<br />
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14.02.2015: <a href="http://simplyphy.blogspot.ch/2015/02/astrooonooomyyyyy.html">Astrooonooomyyyyy</a> : Some random thoughts inspired by some astronomical images. </div>
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Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-18391728757506922692015-04-02T07:57:00.000-07:002015-04-04T01:45:11.593-07:00A relatively special post<div style="text-align: justify;">
In the end of the 19-th century, it seemed like Physics was almost complete. Analytical mechanics was practically finalized with the work of Newton, Lagrange, Hamilton, and of course many others, and Maxwell had just written down a complete description of electromagnetism, building upon the work of Gauss, Faraday, Ampere, etc. There appeared to be a few things left to straighten out here and there, a few small outliers that didn't exactly fit the otherwise beautiful theories, but few people expected any big surprises. You won't believe what happened next!<br />
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The beginning of the twentieth century saw the development of two theories - both arising from scratching these dimples of remaining inconsistencies - that completely changed our understanding of the world. This is by now a textbook cliché, but it's also the simple truth, so I'm OK with re-iterating it. </div>
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The first of these theories is Quantum Mechanics, about <a href="http://simplyphy.blogspot.ch/2015/01/to-be-or-to-not-be.html">which</a> <a href="http://simplyphy.blogspot.ch/2015/01/a-trip-to-copenhagen.html">I've</a> <a href="http://simplyphy.blogspot.ch/2015/02/ladder-to-heaven.html">already</a> <a href="http://simplyphy.blogspot.ch/2015/03/jigsaws-cat.html">written</a> (every word is a separate link). The second one is the theory of Relativity, which I'll (start to) explain here. A remark right off the start: there is the Special Theory of Relativity (STR) and the General Theory of Relativity (GTR); the former is much simpler and came <b>first</b>, which might or might not sound strange (it depends on whether you think deduction is a more natural train of thought than induction). This post is about the STR only.<br />
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The story of STR - and how Einstein came up with it - follows, in a nutshell, the same pattern that has brought some of the greatest insights in the history of science: someone refusing to take for granted something which is obvious for everybody else.<br />
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In the case of Einstein, the statement was, 'obviously, if I see you moving with velocity v<sub>1</sub>, and I see him moving with velocity v<sub>2</sub>, then you see him moving with velocity v<sub>2</sub> - v<sub>1</sub>'. This 'obvious truth' has a slightly more complicated formulation than the ones above (it involves <b>math</b>!), but I'm sure that anybody would agree that it does seem pretty obvious if we only restrict our thinking to <i>practically everything we experience in our lives, ever</i>. But Einstein was totally, like, 'Hmmmm..."<br />
<br />
Of course, he wasn't just toying with an alternative idea out of an abstract <a href="http://whatif.xkcd.com/">what-if-like</a> curiosity. To abandon such 'truths' (postulates), which are deeply in-grained in our thinking, we usually need a very strong nudge. In the case of the STR, this came from the fact that Maxwell had derived an equation for the speed of light, and it had turned out to be a constant - without assuming any particular reference frame! In other words, you assume you're sitting on Earth, you write Maxwell's equations, and you get that you should see light moving with a velocity <i>c. </i>But if you assume that you're moving with velocity <i>c </i>with respect to the Earth, and you write the equations, you get again that light should be moving with velocity <i>c</i>! Initially, people just thought that this is some strange feature of electromagnetism (light is just electromagnetic waves, by the way). They thus tried to propose various extensions to the 'classical' theory, but as experiments kept proving them wrong, the add-ons themselves became more and more convoluted, even for philistine physicists who wouldn't usually care about aesthetics in equations. Einstein always cared about the aesthetics of Physics, by the way.<br />
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Maybe this is the reason why he realized there is an alternative way: instead of adding ever-more-unlikely complications, <b>remove</b><i> </i>just one assumption, even though it seems completely and obviously true - but who knows. So instead of taking the addition of velocities as a postulate, he took a new one: 'light propagates with a constant velocity regardless of the reference frame' (obviously, you cannot have both of those true at the same time). Then, he set out to derive the new laws describing how to go from one reference frame to the other. This transformed the universality of <i>c</i> from a peculiarity of electromagnetism into a fundamental property of space-time. A clarification: by talking about space-time, I'm not trying to sound all sci-fi-y. Instead, this only refers to the fact that if we want to describe any event happening in the world, we have to say where and when it happens. The property of events of having a 'where' and a 'when' makes them embedded in some measure of space and some measure of time, which, combined, we call space-time. Furthermore, I should define more rigorously what a 'reference frame' is: it is the 'where' and the 'when' of the world - so, the space-time - as seen from some particular standpoint. But a standpoint need not be <i>stand</i>ing with respect to other standpoints, and in fact the relationship between <i>moving</i> standpoints is what the STR is all about. Simple, huh?<br />
<br />
I'd like to highlight what I think are the most important aspects of the theory.<br />
1. Einstein assumed just two postulates: that <i>c </i>is constant in all reference frames, and that the laws of Physics are the same in all reference frames (after having been properly translated). In other words, there is a fixed relationship between what I see and what you see, regardless of whether we're moving with respect to one another.<br />
2. Given those two assumptions, he derived the way in which I can translate what I see to match exactly what you see. The theory of relativity, in its purest form, is nothing but a prescription of how to describe an observation to someone who is moving (<i>fast!</i>) with respect to you, in a way that would agree with the other person's observations.<br />
3. All the crazy effects you might have heard of - like space contraction and time dilation, or the new law of addition of velocities (a modified version of the v<sub>2</sub> - v<sub>1 </sub>above that complies with <i>c - c = c</i>) - as well as the not-so crazy but super-fundamental ones - like<i> E = mc<sup>2</sup></i> - can be <b>derived</b><i> </i>based on those two postulates (and the other standard postulates of Physics, like energy conservation, and, you know... mathematics). I always found it truly amazing how much one can achieve with so little to start with.<br />
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<span lang="EN-US"><span style="font-family: inherit;">In the
remainder of this post, I’ll focus on the effects of length contraction and
time dilation. These refer to the fact that different observers might disagree on the distance between two points, or on how much time passes between two events. That is to say, they would disagree as long as they don't use STR to 'translate' their observations in a way that makes them agree. These effects can be derived in many ways; below, I present one illustration of why something fishy has to happen either space or time - or both - if light is to move with the same velocity in different reference frames. Assume that a beam of light travels between two mirrors. If we think about the distance that light covers in a round-trip, there's obviously some difference to be observed depending on whether we are moving with respect to the system (which is equivalent to the system moving with respect to us). <o:p></o:p></span></span></div>
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<span lang="EN-US"><span style="font-family: inherit;">Most generally, in the first case, light travels some distance 2L in some time t, and in the second case - a distance 2L' in time t'. Our intuition would strongly suggest that t' should be the same as t, and that L' should <b>not </b>be the same as L. But if you think about it, this couldn't possibly be the case! Light has to travel with the same speed in both systems, and speed is just distance over time. So 2L/t has to equal 2L'/t', which implies that either we're wrong about t = t', or we're wrong about L' not equal to L, or we're wrong about both... </span></span>In that particular case, we're wrong about t = t', but this is a detail. The big picture implied by STR is that both distances and time periods vary from one observer to the other. These are in fact two manifestations of the same, more general effect: <b>space-time </b>itself is different for different observers. </div>
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Now, t<span style="font-family: inherit;">he
illustration above gives a good intuition of why something fishy must happen when we think of light propagation, but the interesting twist is that, since something fishy actually happens to space and time itself, <b>all</b> physical processes pass on
different time-scales in different reference frames. This, by extension, also
includes chemical and biological processes, and, ultimately, what we perceive
as the passing of time in any meaningful definition (also, aging). This has
been experimentally verified in various experiments. <a href="http://en.wikipedia.org/wiki/Time_dilation_of_moving_particles">One of the seminal ones</a>
involves the decay of particles called 'muons' (they’re kinda like electrons,
but heavier) created by cosmic rays in the top layers of the Earth’s
atmosphere.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHRrD7lMRhBnukRleN4dE_qv8RVYPfU2tUCj2TanqfDHo-lc1qGLr91_PPlcr34pEFlX8m6OoD963R5Swo2jxb4KF0zQFZkc6c3P1tg2jLnJEQlwCuFjGrgT_Y-KMTg79QWw9l1ocQKxE/s1600/muon_creation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHRrD7lMRhBnukRleN4dE_qv8RVYPfU2tUCj2TanqfDHo-lc1qGLr91_PPlcr34pEFlX8m6OoD963R5Swo2jxb4KF0zQFZkc6c3P1tg2jLnJEQlwCuFjGrgT_Y-KMTg79QWw9l1ocQKxE/s1600/muon_creation.png" height="225" title="Nothing is to scale with anything in this image, in no reference frame. " width="400" /></a></div>
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<span lang="EN-US"><span style="font-family: inherit;">We
can create muons here on Earth, in a lab, where we can measure their lifetime (how long they 'live' before decaying into other, more stable particles), which turns out to
be quite short. We also get to detect the muons that come from the cosmic rays,
which 'rain down' on us with an enormous speed (close to that of light).
The funny thing is, though, that if we multiply their speed and their expected lifetime, we should get the maximum distance that the muons could travel before decaying, and... it turns out that the cosmic ones shouldn't be able to get to us <b>at all</b>!<b> </b>The distance they can travel is smaller than the width of the atmosphere, if we don't take the STR effects into account. Thus, we
shouldn't be able to observe the cosmic muons here on Earth - yet we do. Fundamentally, these muons are identical to the ones we make in a lab; the difference comes only from the motion with a 'relativistic' velocity. <o:p></o:p></span></span></div>
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<span lang="EN-US"><span style="font-family: inherit;">This is
a great experiment to demonstrate why time dilation and length contraction are
two different manifestations of the same<b>
</b>effect. The fact that the muons do reach the Earth can be explained in two
different ways, depending on whether you want to stay in a frame in which the
Earth stands still (which you are probably used to</span></span><sup>[<span style="color: blue;"><i>citation needed</i></span>]</sup><span style="font-family: inherit;">), or in one where the muon stays still (remember that the
Physics has to be the same in those two frames, namely, in both of them the
muons have to reach the surface). Let's first have a look at the less intuitive
frame: the muon is standing still, minding its own business, and the Earth is
moving up towards it with a speed close to </span><i style="font-family: inherit;">c</i><span style="font-family: inherit;">.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVCLCzhpcnRnX_nHkqco8IcBGVhfTXu3EJX_gjv-EhHuw0GvKagQkOjaXPWA5hQb_EggUxg_OPri2LwR8rGDC_ZXDO6gBcv67x0EyoXhKzwVK8m78zgDQ51AByT_j7zuFfDVNL1fNLo4s/s1600/earth_moving.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVCLCzhpcnRnX_nHkqco8IcBGVhfTXu3EJX_gjv-EhHuw0GvKagQkOjaXPWA5hQb_EggUxg_OPri2LwR8rGDC_ZXDO6gBcv67x0EyoXhKzwVK8m78zgDQ51AByT_j7zuFfDVNL1fNLo4s/s1600/earth_moving.png" height="201" title="I wonder what kind of literature muons are into" width="400" /></a></div>
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<span lang="EN-US" style="line-height: 115%;"><span style="font-family: inherit;">The life-time
of the muon in this reference frame is exactly the one that we measure for muons
created on Earth; however, since the planet, together with its atmosphere, is
moving towards the particle, the atmosphere appears shorter than its length as
measured on Earth. And so, the particle reaches the surface – or actually, in
this case, the surface reaches the particle! <o:p></o:p></span></span></div>
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<span lang="EN-US" style="line-height: 115%;"><span style="font-family: inherit;"><br /></span></span></div>
</div>
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<div style="text-align: justify;">
<span lang="EN-US" style="line-height: 115%;"><span style="font-family: inherit;">We can also
look at the case where the Earth is standing still, minding its own business,
and the muon is free-falling. <o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKOb1ztbBfxXVUc6ABeaqUdLUoYNRpoZBajBy-ScvcmkBAzdCpQ1cO7D8_aMcfvwnS5IZCurSl4Ia7Od49qfVZKwxgkurXEPRxN8FHNdVXBIkcBiQLuZFFJWOrJbWom5v6L4GnB3OWwF8/s1600/muon_moving.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKOb1ztbBfxXVUc6ABeaqUdLUoYNRpoZBajBy-ScvcmkBAzdCpQ1cO7D8_aMcfvwnS5IZCurSl4Ia7Od49qfVZKwxgkurXEPRxN8FHNdVXBIkcBiQLuZFFJWOrJbWom5v6L4GnB3OWwF8/s1600/muon_moving.png" height="191" title="The glasses are to scale with the book!" width="400" /></a></div>
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<span lang="EN-US" style="line-height: 115%;"><span style="font-family: inherit;">In this case, the length of the atmosphere is
the one we are used to, but the lifetime of the muon is dilated, since it is
moving fast with respect to us – so it has more time to traverse the atmosphere
and reach the surface. And so, as expected, the laws of Physics are preserved.<o:p></o:p></span></span></div>
</div>
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<div style="text-align: justify;">
<span style="font-family: inherit;"><span lang="EN-US" style="line-height: 115%;">Many people,
when they first learn about the relativity of length and time intervals, are
tempted to ask questions like, 'What happens exactly? Does the atmosphere <b>actually </b>get thinner? Does time <b>actually </b>pass more slowly?' The problem
with these questions is that the whole point of the theory of relativity is that there is no 'actually' - hence the name of the theory! There is no 'absolute' series of events in the Universe - they are always described <b>relatively </b>to a given reference frame. In some reference frames, some lengths appear shorter; in others, time seems to pass by more slowly; but in the end, everybody agrees that the muons reach the surface of the planet, one way or another. </span></span></div>
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P.S. There is some polemic about Einstein's role in the whole matter, and how fair it is for him to get <b>all </b>the credit, which he often does. Here's my opinion: it's true that a lot had already been done - the constant speed of light is within Maxwell's equations, and Lorentz had derived the famous transformations that also imply time dilation and length contraction, but it really was Einstein who realized that these two effects are not some strange, minor aspect of electromagnetism, but instead carry a fundamental implication about the very nature of the world. This is as far as STR is concerned; regarding the GTR, Einstein deserves even <b>more </b>credit. So, no, I don't think he's overrated.</div>
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Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-38003762437229597102015-03-08T12:45:00.000-07:002015-03-08T12:45:33.185-07:00Jigsaw's cat<div class="western" lang="fr-CH" style="margin-bottom: 0.14in; text-align: justify;">
<span lang="en-US">Quantum
mechanics written in terms of the wave function and the Schroedinger
equation is quite neat. It is only when we include the 'measurement'
component that things usually <a href="http://simplyphy.blogspot.ch/2015/01/to-be-or-to-not-be.html">become messy</a>, and people start getting lost in 'paradoxes'. But measurements are a part of the <i>postulates </i>of the <a href="http://simplyphy.blogspot.ch/2015/01/a-trip-to-copenhagen.html">Copenhagen interpretation</a>, thus they play a central, unavoidable role in the theory. The conceptual difficulties that this poses are collectively labeled the</span> 'measurement
problem', which represents - I think - the main reason for people to look for
interpretations beyond Copenhagen. I'll try to outline the problem here. </div>
<div class="western" lang="fr-CH" style="margin-bottom: 0.14in; text-align: justify;">
<span lang="en-US">To
do that, let's use the good ol' Schroedinger cat. The setup is the
following: we take a box and put inside a cat and a vial of poisonous gas.
We also put a quantum particle which has a probability of ½ to decay
within one hour, as well as a detector and some mechanism that breaks
the vial as soon as a decay is detected. </span><br />
<span lang="en-US"><br /></span>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQvk1XGciT757D7a3wWzgozL1nPoFJRlHQguF1uXC17O3UBHIwlSFGmiT7m-8O6J6bgjGd6pm8uk0XW1IEDT-n5ZP5oud71p2Qe2Mbn6RKHOcEV4xcL-M1hgDhT0hoCTWHrV3cKtuAeII/s1600/cat_setup.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQvk1XGciT757D7a3wWzgozL1nPoFJRlHQguF1uXC17O3UBHIwlSFGmiT7m-8O6J6bgjGd6pm8uk0XW1IEDT-n5ZP5oud71p2Qe2Mbn6RKHOcEV4xcL-M1hgDhT0hoCTWHrV3cKtuAeII/s1600/cat_setup.png" height="132" title="Animal rights weren't a big thing in the 30s" width="400" /></a></div>
<br />
<span lang="en-US">Thus, quantum
mechanics seems to say that, after one hour, the particle is in a superposition state
of being decayed and not decayed, and the cat is therefore in a
superposition state of being dead and alive. It is at this point
important to note that when Schroedinger proposed this famous thought
experiment, he did not give it as an </span><span lang="en-US"><span style="font-style: normal;"><b>illustration
</b></span></span><span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;">of
quantum mechanics. Instead, he was demonstrating how there is
something </span></span></span><span lang="en-US"><span style="font-style: normal;"><b>wrong
</b></span></span><span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;">with
the way we think of it, because of the nonsense zombie-cat result. He
was pointing out a problem, not giving a solution. This is very important and from now on you should never let anyone get away with saying that Schroedinger's cat </span><b>is </b><span style="font-weight: normal;">simultaneously dead and alive, as if that's some strange paradox of QM that one has to accept as true. Point out their mistake! Be that geeky 'technically, you're wrong' guy or gal! Together, we can start a revolution! And make the world a better place! Yes we can!</span></span></span><br />
<br />
Uhm, anyway. To better understand the measurement problem, let's take a small modification of the thought experiment: let's put a scientist in the box instead of the cat. Let's also put the particle and the vial of poison inside another box. And let's say the scientist inside the
first box is to open the second box one hour from the start of the
experiment</div>
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<br /></div>
<div class="western" lang="fr-CH" style="margin-bottom: 0.14in; text-align: justify;">
<span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;">Ok, let's say the box will open on its own one hour from the start of the
experiment.</span></span></span></div>
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<span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;"><br /></span></span></span>
<span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;">And let's say there's another scientist outside the big box.</span></span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj535HY5kBaiRiGmg3BYp4xncoKO1Ktxsrau_Blu9WQp3w7DAOo-DHw0t_InLsmRN_6XTVib5-inCdgbqU2gjSYb5aNQuFvT7UVH0GkYgHkvFucZrQwjxLX7K_uVS2ZcF_B5CoUKZAFIJ4/s1600/setup_3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj535HY5kBaiRiGmg3BYp4xncoKO1Ktxsrau_Blu9WQp3w7DAOo-DHw0t_InLsmRN_6XTVib5-inCdgbqU2gjSYb5aNQuFvT7UVH0GkYgHkvFucZrQwjxLX7K_uVS2ZcF_B5CoUKZAFIJ4/s1600/setup_3.png" height="202" width="400" /></a></div>
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who's creepily happy about the whole situation<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJP9IuboUMa21XDn8hLhj4B-GsJJ-WZ1_Ihj0ewDN8D1x6W0ODK4rtmhKrmydwc8x6K5WzKy_sJjakUkqqbe6P0l0IjP5oViJLclusSk5cMKnWvolx258KjaW1X5sPOolqrk82tjThg-I/s1600/setup_4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJP9IuboUMa21XDn8hLhj4B-GsJJ-WZ1_Ihj0ewDN8D1x6W0ODK4rtmhKrmydwc8x6K5WzKy_sJjakUkqqbe6P0l0IjP5oViJLclusSk5cMKnWvolx258KjaW1X5sPOolqrk82tjThg-I/s1600/setup_4.png" height="193" width="400" /></a></div>
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because it's actually a cat controlling a humanoid robot!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsQjKfdAoxE3SfwwGjbBo-jPGPb3-5Y2uX2ph7jxlYjF8mFf4Me4iJgUj2h_50Nk7TQK_xENqKXB8LPcsW-ZfrvGDpVYeNDO-lt3o9uNeML7KkLuCVfJ5pobbzJfu0Zdbh8P7Hm1-aItg/s1600/setup_5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsQjKfdAoxE3SfwwGjbBo-jPGPb3-5Y2uX2ph7jxlYjF8mFf4Me4iJgUj2h_50Nk7TQK_xENqKXB8LPcsW-ZfrvGDpVYeNDO-lt3o9uNeML7KkLuCVfJ5pobbzJfu0Zdbh8P7Hm1-aItg/s1600/setup_5.png" height="236" width="400" /></a></div>
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<span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;">It seems that in this situation it is extremely difficult to say at which point of time a measurement occurs and the particle's wave function 'collapses'. Say
that the detector measures a decay already at the fifth minute. Has the wave function collapsed already? As far as both scientists are
concerned, no it's not, because they haven't 'measured' it yet – they
don't know whether the decay has happened. On the other hand, some sort of measurement did take place: the apparatus interacted with the system, right? So it must have also broken the vial of poison. Then, when 55 minutes more pass and the box opens, the second scientist is dead. Was that a 'measurement'? Is the wave-function collapsed now? As far as the cat-scientist is concerned, no it's not, because he hasn't 'measured' it: the big box is still closed. Or, let's have
mercy and remove the vial of poison. When the small box opens, the
second scientist can simply read out what the detector says </span></span></span>
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<span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;">Has the wave function of the particle collapsed then? The detector has
now been read out, so maybe yes, but again that's only the case from the
point of view of the scientist in the box. As far as the cat outside is
concerned, the system is still in a superposition state, and if he didn't
know about us removing the vial of poison, the second scientist is
himself in a superposition state of being dead and alive (Schroedinger's original nonsense result). Can we say that the wave function has collapsed because a conscious being has read out a detector? That's obviously a very dangerous path to take. What is consciousness anyway? Is a cat a conscious being? What if it's smart enough to build a controllable humanoid robot? </span></span></span></div>
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<span lang="en-US"><span style="font-style: normal;"><span style="font-weight: normal;">Bottom
line, the measurement postulates present many difficulties on the philosophical level. When does the measurement happen? Does it happen
instantaneously? This is a huge problem in cases in which it would imply that an action at one point of space has an instantaneous effect at a
different, distant point: that could break the idea that there is a cause-effect relationship in nature. Anyway, how do we even split the world into 'quantum'
and 'classical' – isn't everything inherently quantum? That we are able to do such a split is pre-supposed in the Copenhagen postulates, but no rigorous way to do that is given. What if we want to study the whole universe as a quantum system?</span></span></span></div>
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In case that hasn't convinced you yet that there is a problem, I'll now return to the original problem as posed by Schrodinger. There is actually more than one way to 'resolve' it within the Copenhagen interpretation. <a href="http://simplyphy.blogspot.ch/2015/01/to-be-or-to-not-be.html">Here</a> I suggested one: do not ascribe any classical properties to the cat before it is observed - it's not a cat, it's a wave function in a superposition state. But it is also possible to think of the cat as a nice, ordinary, classical kitty, which has some probability to be dead, but is <i>not</i> in a superposition state (because that wouldn't make any sense, or would it?) In other words, since there is no rigorous definition of how to split the world into quantum and classical, it's up to us to decide to which world the kitty belongs. What's the difference between the two representations of the situation? I hope this makes it clear:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOHDyAcziUiBKk5uWV4rJc160Lk0ATPXR4akLiTblOi2d2-BqZTRvqu7YWOCGtfQWpgHb90FhSgWz4O0uYPW5GR4rX3LF7tFk6GG_-SsmTA3o4GYOgGlUl6KK-_I4Y1IDhc-DGu_3loQ0/s1600/cat_version1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOHDyAcziUiBKk5uWV4rJc160Lk0ATPXR4akLiTblOi2d2-BqZTRvqu7YWOCGtfQWpgHb90FhSgWz4O0uYPW5GR4rX3LF7tFk6GG_-SsmTA3o4GYOgGlUl6KK-_I4Y1IDhc-DGu_3loQ0/s1600/cat_version1.png" height="130" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgO3qpkKCql43Jq4VJawdUPP5-ITuh-Ct4Er6vdoHRealhjTNNvzPObXan-j6XkIxwR1cSxQ5VeNwLNymQDQ85oTfjgwK29eQJF_ZKSqmgSWFZanMzlUkOZdkS2gvRtXFUmZVV4XCQAJg4/s1600/cat_versions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgO3qpkKCql43Jq4VJawdUPP5-ITuh-Ct4Er6vdoHRealhjTNNvzPObXan-j6XkIxwR1cSxQ5VeNwLNymQDQ85oTfjgwK29eQJF_ZKSqmgSWFZanMzlUkOZdkS2gvRtXFUmZVV4XCQAJg4/s1600/cat_versions.png" height="202" width="400" /></a></div>
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Different people might prefer one or the other version. I think Schroedinger had the first version in mind when he proposed the problem. In the second version, it's fun to think of the cat as a classical measurement apparatus which is constantly measuring the wave function of the quantum particle... by dying or not dying! In that sense, the wave function cannot be in a superposition state, it's either the wave function of a decayed particle or of an un-decayed one, because at every moment the cat interacts with it and 'measures' which is the case, causing an immediate collapse.<br />
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The problem is not with the versions themselves as much as with the fact that they are both allowed. The problem is, in other words, with the supposition - inherent to the Copenhagen postulates - that we can divide the universe into a quantum part, which is described by quantum mechanics, and a classical part, in which we live. Cats, too - or do they? What if instead of a cat we had a bacteria? Should we include it in the quantum system or keep thinking of it classically? What if we had a molecule that would dissolve if exposed to the gas? Is the molecule 'quantum' enough for us to have to include it in the wave function and not in the classical world? And if yes, where do we draw the line between a molecule and a cat... and is this line different for every different setup?<br />
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At this point, people who really want to stick to the Copenhagen interpretation will inevitably start talking about decoherence, which is a complex topic, but which, to me personally, always leaves a tang of not really solving the problem. In other words, I think that everybody who is sufficiently bothered by the problem to feel the need to dig deeper has to inevitably leave Copenhagen in search of further answers. And, of course, I'm planning to get to that one day!</div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-77476271549093930002015-02-14T12:18:00.000-08:002015-02-14T12:18:48.517-08:00Astrooonooomyyyyy<div class="separator" style="clear: both; text-align: center;">
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The title (and the multiplicity of vowels therein) is inspired by a hugely unknown Metallica song from the Garage Inc. album called... well, 'Astronomy' - which is, like everything on the album, a cover. The original song is by Blue Öyster Cult. Do check it out (both versions are pretty cool)!</div>
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So, to illustrate that I won't be talking only about quantum mechanics here, let me share some random thoughts about some astronomical images. Like, maan, nebulas are pretty:</div>
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<a href="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Orion_Nebula_-_Hubble_2006_mosaic_18000.jpg/768px-Orion_Nebula_-_Hubble_2006_mosaic_18000.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Orion_Nebula_-_Hubble_2006_mosaic_18000.jpg/768px-Orion_Nebula_-_Hubble_2006_mosaic_18000.jpg" height="400" width="400" /></a></div>
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This one is a vary famous one, the Orion nebula. Credit for the image goes to the Hubble Space Telescope, which has in fact done an amazing job: the actual resolution of the image is 18,000 by 18,000 pixels, and a fun zoomable version is available <a href="http://hubblesite.org/newscenter/archive/releases/2006/01/image/a/format/zoom/">here</a>. Until recently I had never thought about how big nebulas actually are, but I know that they are formed after a star burns out and explodes into a supernova, so I thought that gave me a pretty good idea. They turned out to be slightly bigger than that idea, but not too much. Well, although I cannot say that they are huge by astronomical standards (insert a 'your mom' joke here), they certainly are pretty huge for something created by a single star. Let's stop for a second and admire another one. </div>
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<a href="http://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/768px-Crab_Nebula.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/768px-Crab_Nebula.jpg" height="400" width="400" /></a></div>
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That's the Crab nebula. Credit goes again to Hubble, and again there is a <a href="http://hubblesite.org/newscenter/archive/releases/2005/37/image/a/format/zoom/">zoomable version</a>. Pretty amazing, huh? Let's see how big it is. Let's start from the standard comparison, Earth, Sun, distance between the two.</div>
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On the left side, the dot representing the Earth is to scale compared to the ball that is the Sun; on the right side, the dot representing the Sun is to scale with the Earth-Sun distance, but the dot representing the Earth is obviously not. The distance between the Earth and the Sun is a unit of length commonly used in astronomy, and as such, it's rather appropriately named... 'astronomical unit'. 1AU is about 150 million kilometers. And here is how this distance compares to the rest of our Solar System (and a bit beyond! Have you heard that <a href="http://en.wikipedia.org/wiki/Voyager_1">Voyager 1</a> not long ago became the first mam-made object to leave our star system?)</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRzbWDfAHFnvjl9-TGXIqaImXnoqI3YPRNvH2yyjsnuja9l-PYHvvSy3m0YspUmiff98ePvhc4o2zR4qA_er-L1_1MoF_i-fgJh5pETLnS3Yp0QrCLPGy0QiK41yH37pIECBlcdUnhL6I/s1600/au-vs-solar-system.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRzbWDfAHFnvjl9-TGXIqaImXnoqI3YPRNvH2yyjsnuja9l-PYHvvSy3m0YspUmiff98ePvhc4o2zR4qA_er-L1_1MoF_i-fgJh5pETLnS3Yp0QrCLPGy0QiK41yH37pIECBlcdUnhL6I/s1600/au-vs-solar-system.png" height="273" width="540" /></a></div>
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We're however still far from the required length-scale to describe the size of a typical nebula, i.e. they are much larger than the entire Solar System! I cannot come up with anything familiar whose size is in between the Solar System and the nebula size I am trying to get to. Also, I seem to have an unexpected problem in writing this post in that I want to put a 'your mom' joke in almost every paragraph. Resisting the temptation, let me define the size of the Solar System (twice the distance from the sun to the heliopause) as a unit of length, for which I'll use the (unfortunate) abbreviation SS. For people more familiar with standard internet units of measure, 1SS is approximately 179.51744 × 10<sup>12</sup> bananas, although it depends where you buy them from. That's 179.5 trillion, for the people unfortunate enough to not understand scientific notation. </div>
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Ok, so having defined that, we can define one Deca-SS (10 times SS), and one Hecto-SS (you guessed it!).</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUVLy3Tv8Cwj4Aq4ptCPkpzyOMwsK6RQRdH46mFYjJp2fYyoEfR1JI2YjhWTqLXIaImvLiJjFGDD34_6iixhLK7KGLRswV97T9OAi6C0rtja0reyFCDv6rKE7YiWHRqwIjtkYBZUsAQz8/s1600/SS-s.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUVLy3Tv8Cwj4Aq4ptCPkpzyOMwsK6RQRdH46mFYjJp2fYyoEfR1JI2YjhWTqLXIaImvLiJjFGDD34_6iixhLK7KGLRswV97T9OAi6C0rtja0reyFCDv6rKE7YiWHRqwIjtkYBZUsAQz8/s1600/SS-s.png" height="150" width="400" /></a></div>
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The Hecto-SS is a unit that's finally big enough to be visible on the scale of the Crab nebula.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi24GmJlR4AP5R1RjTZQVKxsEUTzPUo5GYKos6LN_mODhTYdkr_cLvigp-nROOGEoJOFHYkPafO8k0eq4XFEBWXEGBAyJjffb6wGXOoNUmuaPVU5QNCGYJ6ZX0IhXyj4L6PCIUOv_fHLig/s1600/crab-ss.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi24GmJlR4AP5R1RjTZQVKxsEUTzPUo5GYKos6LN_mODhTYdkr_cLvigp-nROOGEoJOFHYkPafO8k0eq4XFEBWXEGBAyJjffb6wGXOoNUmuaPVU5QNCGYJ6ZX0IhXyj4L6PCIUOv_fHLig/s1600/crab-ss.png" height="247" width="400" /></a></div>
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More precisely, the nebula is about three thousand times bigger than our entire Solar System! So, yeah, pretty big, huh? Or maybe you're disappointed, cause you expected it to be bigger? (zing!) Anyway, with astronomical distances it's really hard to know what to expect, but now you do, in case you ever <a href="https://www.youtube.com/watch?v=dsw3TILXXBk">want to be a millionaire</a> (and by incredible chance you get asked just that). </div>
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Now, what got me thinking about nebulas in the first place was not their size. Instead, I was wondering if the above images are just pretty pictures, or if they are a representation of reality. This is because scientific images which are taken using some apparatus, like a telescope or a microscope, are sometimes in 'fake color', which means that the apparatus does some coloring following some rules. There is usually a very good reason to do that, and the rules usually encode some information in the colors. So what about the nebulas, could we ever hope to see them in all their majesty (i.e. as in the images above) with the unaided eye? Not exactly. First of all, to see the nebulas as big as in the images, we would definitely need some aid: either a telescope, or, if we have somehow gotten close enough to need no magnification - a filter, because the light will be too bright. But in addition to that, for nebulas, Hubble does use a color-coding: namely, it assigns a particular color to a particular chemical element. For example, in the image of the Crab nebula, blue represents neutral oxygen, green indicates singly-ionized sulfur, and red indicates doubly-ionized oxygen. </div>
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So, then, are those just pretty images, or are they also an accurate portrayal of reality? I would say that they are the latter. What <i>is</i> reality anyway - we definitely cannot restrict the definition to whatever we can see with our eyes. And in a way, in the Hubble images reality is portrayed even better, because, while the nebulas will no doubt still look stunning unmodified, this color-coding allows us to differentiate the constituent elements even better. In other words, as long as the image conveys scientifically accurate information, to me it is indeed a representation of reality, regardless of what part of it our eyes could see if unmodified. Take as another example this stunning image of the Milky Way</div>
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<a href="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/ESO-VLT-Laser-phot-33a-07.jpg/1024px-ESO-VLT-Laser-phot-33a-07.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/ESO-VLT-Laser-phot-33a-07.jpg/1024px-ESO-VLT-Laser-phot-33a-07.jpg" height="265" width="400" /></a></div>
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This is reality right there, it's something that is always just above our heads, ready to be gazed at, but we can never hope to see it like that (unless we teach our brains to take long exposure shots...) But still, this awesomeness exists above us and it is only our sensory limitations that prevent us from appreciating it. Fortunately, however, we are getting better and better at using tools to capture reality, and represent it in an accurate way that our crippled senses could actually appreciate. Great stuff. </div>
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But this is not always the case: there is another type of 'fake' images which sometimes appear especially in relation to astronomy, and that brings me to the original inspiration for this post. This image of a recently discovered exoplanet became mildly viral in the last weeks</div>
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<a href="http://cnet1.cbsistatic.com/hub/i/r/2014/04/17/4e7441aa-945c-4f78-92e8-99bcf2582f08/resize/770x578/ebbfa1af42e68bb55c616974ab249f07/kepler186f.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://cnet1.cbsistatic.com/hub/i/r/2014/04/17/4e7441aa-945c-4f78-92e8-99bcf2582f08/resize/770x578/ebbfa1af42e68bb55c616974ab249f07/kepler186f.jpg" height="179" width="320" /></a></div>
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Now, the important thing to realize here is that everything we know about Kepler-186f is inferred through observations of the star it orbits; we have no way of capturing any image of the planet itself, let alone one as detailed as the one in the picture. What's nevertheless pretty cool is that just by observing the star, we can infer with quite a lot of certainty that there has to be a planet orbiting it, and, furthermore, we can estimate the size and orbital radius of the planet. Based on this, we know that around a certain star there orbits a planet that's slightly larger and slightly colder than the Earth, but that's literally everything we know with certainty. We don't even know its mass, so we can only guess what it could be made out of, and we certainly have absolutely no idea if there's water or not. So the planet on the left is just someone's imagination of what a slightly larger, slightly colder Earth could look like - one of the infinite number of possibilities. Unlike the Hubble images, there is nothing scientifically accurate in the details of this picture. Images of this type are called 'artist's conception'. Don't get me wrong, I'm not calling the image <i>fake </i>or anything, and I'm not against images like this, but I do think that if they belong to that type, that should be made very, very clear, because a lot of people could be fooled. </div>
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To conclude, since I mentioned the Milky Way earlier, we all know and love our home galaxy, right? </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA7ut5SZyUxicEbvNUvAP41KKCMDCooK8927cMaVC9qBdKfGTHOB-p-Kk_nv-hmt8hIut1shmwyRwbH53UoFJaMCcx9cBF2fONbMxy5CT6LZFLjZrJVwilK3swdLioo0K_MaLRBo_2VhU/s1600/713px-Wide_Field_Imager_view_of_a_Milky_Way_look-alike_NGC_6744.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA7ut5SZyUxicEbvNUvAP41KKCMDCooK8927cMaVC9qBdKfGTHOB-p-Kk_nv-hmt8hIut1shmwyRwbH53UoFJaMCcx9cBF2fONbMxy5CT6LZFLjZrJVwilK3swdLioo0K_MaLRBo_2VhU/s1600/713px-Wide_Field_Imager_view_of_a_Milky_Way_look-alike_NGC_6744.jpg" height="336" width="400" /></a></div>
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But this is not the Milky Way! That's another galaxy (NGC 6744), and we just think that the Milky Way should look something like this. It took me an embarrassingly long time to realize that we cannot take an 'outside' picture of the Milky Way like the one above, cause we are hopelessly trapped on the inside of it. To put it simply, we don't have a giant, inter-galactic selfie stick.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFzGLknV5iztzMuoVxaZLjtt6yWzCSB-D0kg7O0al_F6HNWW3_70_mCf348UxkZTqBuRghUnhNYS5MLpKN0JV2Nru1NsV4sSZnuech1WYRcWRL4pu3BVxeA05PbkyB53On5JX5KTjdKtM/s1600/selfie.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFzGLknV5iztzMuoVxaZLjtt6yWzCSB-D0kg7O0al_F6HNWW3_70_mCf348UxkZTqBuRghUnhNYS5MLpKN0JV2Nru1NsV4sSZnuech1WYRcWRL4pu3BVxeA05PbkyB53On5JX5KTjdKtM/s1600/selfie.png" height="283" title="'Inter-galactic selfie stick' sounds pretty cool. NASA? ESA? Anyone?" width="400" /></a></div>
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<br />Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com1tag:blogger.com,1999:blog-6550459644052915218.post-56337220469532424722015-02-01T07:25:00.001-08:002015-03-11T00:22:12.185-07:00Ladder to heaven<div style="text-align: justify;">
I hope I've made it clear how within (<a href="http://simplyphy.blogspot.ch/2015/01/a-trip-to-copenhagen.html">my interpretation of</a>) the Copenhagen interpretation one doesn't really <b>need</b> the notion of a 'particle'... Cause now I will go back to using the word 'particle' occasionally, in a very general sense. There are several reasons to do this. There certainly is a semantic one: quantum mechanics describes things like atoms and electrons and photons and so on, and we are used to calling those things by the generic name 'particles'. There's also the physical one, namely the fact that in observations the wave function sometimes shows particle-like properties, like a very well-defined position in space. But I think the semantic reason is stronger - I really don't want to go on writing 'a quantum object' instead of 'a particle' whenever I want to refer to what the wave function describes. However, I do urge you, whenever you see the word 'particle' and are dealing with QM, do <b>not</b> imagine a hard, solid ball shooting around through space: the wave function obviously has little to do with that. On the contrary, it is in many ways like a standard wave - like on the surface of a pond, for example - but has one additional property that ultimately leads to the whole particle/wave fiasco. Let me try to explain it, which in a way will help me put the 'quantum' into quantum mechanics. But before doing that, let's take a look at a guitar string.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgutC7DD8ThcH4WqCM8CS19tWwazQk9_XwOIILv2t3PH7oI2BknFX-P6B2fKqKiM9N4f3OLqK41J-9JJfmke3_V9qECjfU2dQdYhGAjWV-M1EPVotDqtmMBusr5s9-HNzZDu9GNoOVp-X0/s1600/guitar.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgutC7DD8ThcH4WqCM8CS19tWwazQk9_XwOIILv2t3PH7oI2BknFX-P6B2fKqKiM9N4f3OLqK41J-9JJfmke3_V9qECjfU2dQdYhGAjWV-M1EPVotDqtmMBusr5s9-HNzZDu9GNoOVp-X0/s1600/guitar.png" height="141" title="If there happen to be some geeky chicks around that just might lead to putting something else into yet another thing." width="400" /></a></div>
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The tone that a guitar string makes when pinned between two points is determined by, among other things, its length. More precisely, the length determines the wavelength of the possible vibrations of the string, which is inversely proportional to the frequency of vibration, which is what our ears register. Different wavelengths mean different frequencies mean different tones. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhILTUEiJ_ggBHyO9T9MlOd-VqOkLcsPZrj4EzM8Dskz3vz-l82qSqtU7Je5d7iA2lGpdsJ2AcHA11WzZmrd9t6WyxOlypw1Q0h1Q3MTmocqFeXEGKOTApgvv5APSpg3Z7TUB9CXvO_b84/s1600/guitar_tones.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhILTUEiJ_ggBHyO9T9MlOd-VqOkLcsPZrj4EzM8Dskz3vz-l82qSqtU7Je5d7iA2lGpdsJ2AcHA11WzZmrd9t6WyxOlypw1Q0h1Q3MTmocqFeXEGKOTApgvv5APSpg3Z7TUB9CXvO_b84/s1600/guitar_tones.png" height="203" title="That's Eeeee, Ggggg, Bbbbb, if you're wondering." width="400" /></a></div>
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The ‘fundamental’ tone of the string is the one with the longest wavelength, and this is what mostly comes out when we pluck the string, although the other tones are generally also present, and are called '<a href="https://www.youtube.com/watch?v=vC9Qh709gas">overtones</a>'.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnDFetSZT6my7-QqVXGYL6ZTk86EaiN5mxTtc-kQLKdXE5mWv5XHZ06_bMbISFTqQgrbsAi230LNn83OD0GYSXYAY-TpJfPzeS4UyD_4ya6IYS6pY5K680GgKx1j87Yq8o-ZrWrOx695Q/s1600/guitar_overtones.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnDFetSZT6my7-QqVXGYL6ZTk86EaiN5mxTtc-kQLKdXE5mWv5XHZ06_bMbISFTqQgrbsAi230LNn83OD0GYSXYAY-TpJfPzeS4UyD_4ya6IYS6pY5K680GgKx1j87Yq8o-ZrWrOx695Q/s1600/guitar_overtones.png" height="157" title="That's Eeeee, Eeeee (one octave higher), and Bbbbb in that octave, in case you're still wondering!" width="400" /></a></div>
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By pressing on various frets, we make the fundamental wavelength shorter or longer, and produce various tones. So here is how the beginning of Stairway to Heaven goes, if played on one string (well, for the fifth row, which represents two notes played simultaneously, you'll need two identical strings).<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4Fg2jfF3mS7FDGlmIX3JfurP-69qr6LKD1G0WR_2IXL4lq0foL-VysaJphp_uEqrk_bhedyL6HHgUgl_rlLqVzdIULZkNvuVKRBoA-NMpPmORQy9wTI48_LZUvZvNDuSuh5totHE0M2U/s1600/stairway.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4Fg2jfF3mS7FDGlmIX3JfurP-69qr6LKD1G0WR_2IXL4lq0foL-VysaJphp_uEqrk_bhedyL6HHgUgl_rlLqVzdIULZkNvuVKRBoA-NMpPmORQy9wTI48_LZUvZvNDuSuh5totHE0M2U/s1600/stairway.png" height="202" title="La do mi la (sol# + si)" width="320" /></a></div>
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That's actually not at all relevant to my point but I thought it's cool enough to show. My point goes along the lines that any vibration of the string has an associated energy to it. For a given wavelength, the energy is also related to the amplitude of the vibration: the farther the string goes up and down, the more energy is present in its motion. Consequently, more energy is transferred to the air in the form of sound waves, and we hear the tone louder. So there is a continuous relationship between amplitude and energy:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqsFHqLifLL2X0sDvdxTPHCp6lCs0sV-ard3gB3evNYtc2Th-r2okqGfdojB4hEbVh_neWCahQlZmn2jSUaRTwa-I34NzLQsNxM1C_RzkEEeu1eOsLPlgcEBE5DMP-Gavp8CBE6UNB5-A/s1600/guitar_amplitude.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqsFHqLifLL2X0sDvdxTPHCp6lCs0sV-ard3gB3evNYtc2Th-r2okqGfdojB4hEbVh_neWCahQlZmn2jSUaRTwa-I34NzLQsNxM1C_RzkEEeu1eOsLPlgcEBE5DMP-Gavp8CBE6UNB5-A/s1600/guitar_amplitude.png" height="159" title="If you're still wondering, I'm not sure why you're reading this blog in the first place." width="320" /></a></div>
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One of the simplest QM systems to study is called the 'particle in a box', and is very similar to the guitar string. Assume the wave function can only be non-zero between two points in space, but in between those points it is free. This practically corresponds to 'trapping' a quantum particle inside a 'box'. What are the possible solutions to the Schroedinger equation? (i.e., what happens?) Well, it turns out that the allowed 'vibrations' of the wave function look exactly the same as the fundamental tone and the overtones of the guitar string.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjm87fjr29svg9V4rRRWPwxp1Cwbsn-uNkllZa-tVm5qvuzl4NsWgt72sNj6QHGOFSJi7vpM5gal7gAZW8F5oNwAop_qcGGpd_NlSmBlWbSx9XdgsuYSCCcUMXSJZSSaB8gwzQdkmYFpbg/s1600/pib_energies.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjm87fjr29svg9V4rRRWPwxp1Cwbsn-uNkllZa-tVm5qvuzl4NsWgt72sNj6QHGOFSJi7vpM5gal7gAZW8F5oNwAop_qcGGpd_NlSmBlWbSx9XdgsuYSCCcUMXSJZSSaB8gwzQdkmYFpbg/s1600/pib_energies.png" height="158" title="Quantum guitars: the future of music? " width="320" /></a></div>
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But in the case of QM we don't call these 'tones': instead, we directly refer to them by their associated energies, which I've labelled in the figure. So far, the wave function looks like an ordinary wave (which is practically the reason why it's called like that in the first place), but here comes the fundamental difference: for every 'tone' of this 'quantum guitar', the amplitude is fixed! For example, there is exactly one energy E<sub>1</sub> which is associated to the fundamental tone, and we cannot modify the amplitude of the vibration.<br />
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Here's why. The wave function defines a probability density, which stems from the fact that results of measurements can be (probabilistically) predicted by it. Thus, if we call the wave function Ψ(x), then |Ψ(x)|<sup>2</sup> gives the probability of observing a particle at a given position, x. Now, probabilities must always have the following property: when you add them all up for all possible outcomes, the sum must be equal to one (i.e. 100%). This is practically the definition of 'all possible outcomes': if you try to measure the particle at any possible place, you are 100% sure to find it somewhere. This means that the wave function follows a pretty standard wave-like equation, similar to the ones we find in electromagnetism or fluid dynamics, but on top of that there's an extra condition – its absolute value squared has to 'integrate' to one.<br />
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This is what fixes the amplitude. In other words, then, one cannot modify the energy of the tone by 'plucking the string harder'. In fact, if we want to 'pluck' the wave function, that is, to add energy to it, we can only do that by adding exactly the right amount so that we excite a higher-energy overtone. If instead we try to give it an amount of energy which is 'not good', nothing will happen: the wave function cannot 'accept' that, because it cannot use it for anything!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlhjyObg5Pn9BZr-kmrcp8L7T6wpGWq4a-THCtY2WsTRNhdpr7TP7CXuDBlArj_Dm50eNMAgPnyDJ3dFXsOGRlUc4XfR4inGoQFuUAUi_FKbbpEnqWIzHH02rF2-BS4eVOO-UzRmujwEM/s1600/wf_present.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlhjyObg5Pn9BZr-kmrcp8L7T6wpGWq4a-THCtY2WsTRNhdpr7TP7CXuDBlArj_Dm50eNMAgPnyDJ3dFXsOGRlUc4XfR4inGoQFuUAUi_FKbbpEnqWIzHH02rF2-BS4eVOO-UzRmujwEM/s1600/wf_present.png" height="201" title="Is it sexist to think of the wave function as a guy? " width="400" /></a></div>
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The fact that only discrete energies (physicists call them 'energy levels') are allowed is what is called 'energy quantization'. The word 'quantum' basically means a 'packet', or more precisely some quantity which cannot be divided into smaller quantities. The fact that energy comes in 'quanta' is one of the very fundamental properties of QM (hence the name!) - and is always the case for a finite quantum mechanical system (and one can argue that infinite ones are unphysical). By the way, the energy levels of the system, E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, etc. are sometimes called a 'ladder', which is the origin of the admittedly lame-ish pun in the title of this post.<br />
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So there you have it. If you've followed this far, you can brag around that you know what 'quantum' means. It has always struck me as quite funny that many people know a little about quantum mechanics, but very few non-physicists have any idea what the word 'quantum' actually refers to! Of course, I'm not claiming that I have given here a strict, exhaustive definition of the term - not at all, I just gave a simple illustration. But that's practically where the word came from in the first place: people saw that certain systems accept only some very well defined packets of energy, and called them quanta, and... that's it!<br />
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Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-13377977889994770612015-01-30T00:45:00.002-08:002015-02-19T02:22:37.780-08:00A trip to Copenhagen<div style="text-align: justify;">
This is an auxiliary post. It's quite technical and not at all funny, apart from two geeky jokes that I've put in the very end. I was preparing a new 'main post', but I realized it's worth defining the Copenhagen interpretation a bit more rigorously. I hope this might still be of interest to a fairly general audience. Apart from helping me with the next post, this is also strongly related to the previous one, but, again, it is very technical, so don't expect to be entertained. Or, you know, do. You're probably all geeks who get their kicks out of that kind of stuff.</div>
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Here is how I would formulate an essential definition of what is referred to as 'the Copenhagen interpretation'. Notice that there is no single, unanimous definition, so take this as... my interpretation of the interpretation. If you read e.g. wikipedia's statements of the <a href="http://en.wikipedia.org/wiki/Copenhagen_interpretation#Principles">principles</a>, you'll notice some differences. Anyway. I do <b>not </b>claim any formal rigor here, in fact I am trying to put the essential postulates in words as simple as I can find. </div>
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<li>Excluding measurements, a quantum system is fully defined by its wave function, which evolves smoothly in time. We call a particular wave function Ψ a 'state' of the system.</li>
<li>A 'measurement' is the interaction of the quantum system with a properly designed apparatus that is 'classical', i.e. it has properties that make sense in terms of classical mechanics. Let's say that this roughly means that the apparatus presents information that we can directly read out, e.g. in the form of the position of a dial.</li>
<li>A measurement apparatus has an in-built set of mutually exclusive (see postulate 5) states Ψ<sub>1</sub>, Ψ<sub>2</sub>, ..., each of which is an allowed state of the system, such that its operation can be summarized in its asking the wave function: 'which of those states do you want to be in?' The answer is then displayed for us to see, e.g. the dial shows '1' or '2' or whatever.</li>
<li>If a measurement has been made (the dial turns to a given value), the wave function 'collapses' onto the state that the dial shows. This means that regardless of what it used to be, the wave function becomes e.g. Ψ<sub>2</sub> if the dial shows '2'. It then proceeds to evolve smoothly in time until the moment in which another measurement is made.</li>
<li>The probability of measuring the n-th possible outcome is computed as the absolute value squared of what is called the 'overlap' between the initial wave function and the corresponding Ψ<sub>n</sub>. Or in other words: there's a mathematical formula that lets you compute the probability of a particular measurement outcome based on the initial state. The quality of the in-built apparatus states being mutually exclusive refers to their having zero overlap with each other; in other words, if the system right before measurement is already in one of the built-in states Ψ<span style="font-size: 13.3333330154419px;">i</span>, it has a probability of 0 to be measured in any of the other states Ψ<sub>n</sub>, where j is not i.</li>
<li>(This is more a clarification than a postulate.) Before measurement, the wave function need not be in any one of the states 'allowed' by the apparatus; it can be in a superposition of them, or even, it might be in a state which is completely unknown to the apparatus. In this last case, the 'overlap' with each of the in-built states - and thus the probability for a measurement to even take place - is zero, and the system does not interact with the apparatus at all. In mathematical terms, the set of states need not be complete (i.e. including all possible outcomes). </li>
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Often, concepts like the wave-particle duality or the uncertainty principle are discussed at the same time as these fundamental principles, but this is really not needed. I really don't like the wave-particle duality anyway (notice that I never had to use the word 'particle'!), and the uncertainty principle is alright, but it's more of a corollary to the above postulates than a fundamental principle on its own. I'll get to that some day. </div>
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Here are your two geeky jokes:</div>
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1. Schroedinger's cat walks into a bar. And doesn't.</div>
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2. A Buddhist monk at a hod-dog stand says, 'make me one with everything.'</div>
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I like only one of those jokes because the other perpetuates a largely misunderstood concept. If you're not sure what I'm talking about, you can (re)read <a href="http://simplyphy.blogspot.ch/2015/01/to-be-or-to-not-be.html">this</a>. In fact I wanted to put two jokes about quantum mechanics here, but I <a href="http://www.jupiterscientific.org/sciinfo/jokes/physicsjokes.html">skimmed</a> <a href="http://jcdverha.home.xs4all.nl/scijokes/2_10.html">through</a> <a href="http://iopenshell.usc.edu/forum/topic.php?id=20">some</a> <a href="http://www.quora.com/What-are-some-of-the-best-Quantum-Mechanics-jokes">websites</a> and did not find a single one which was both original and did <b>not </b>make me want to be like, 'well, that's not <i>exactly </i>right!' So I decided to go with the Buddhist one, which always makes me chuckle. :)</div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0tag:blogger.com,1999:blog-6550459644052915218.post-50002975315362414972015-01-10T01:01:00.000-08:002015-04-19T06:12:21.589-07:00To be or to not be<div style="text-align: justify;">
<span style="font-family: inherit;">By popular demand,<sup>[<span style="color: blue;"><i>citation needed</i></span>]</sup> here's a place where I'll put down my very own understanding of Physics and discuss some of its (hopefully) interesting aspects. While I do hope that this be of interest to professionals, I imagine they might often find the content obvious. In fact I <b>hope </b>that they find it obvious, as opposed to, say, wrong. On the other hand, I do believe that some of my viewpoints are personal insights rather than canonically taught material (I guess they wouldn't be 'viewpoints' otherwise?), thus if a colleague finds them useful - great, and if they point out where and why I'm wrong - even better.</span></div>
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<span style="font-family: inherit;">For the most part, however, my target audience is laymen with general interest in science. This is not only because I'm not afraid of criticism coming from that side, although I'm not. I've seen media articles getting away with all sorts of inconsistencies. And <b>this</b> is the reason why I'm setting out on a quest to straighten those out. Well, not really, or rather, I'm not really hoping I'll be able to do that. But I do think that a lot of the 'strange' aspects of Physics have been unnecessarily mystified - in some cases because of lack of better understanding, and in others - perhaps more often - because it just sounds cooler that way.</span><br />
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The champion theory in terms of mystification is of course Quantum Mechanics (QM), which is what the remainder of this post is about. Getting a good intuition of the quantum world is indeed <a href="http://abstrusegoose.com/93">quite tricky</a>, which is I guess why it has come to serve as an excuse for scientists and laymen alike to <a href="http://en.wikipedia.org/wiki/Quantum_mysticism">discuss metaphysical phenomena</a> like the soul, our place in the world, or telepathy, in a tone that sounds scientific but is most often not much. What's annoying is that I think it <b>is</b> possible to see the quantum world in an intuitive way, with a little bit of effort - and its 'strangeness' should not be taken for granted. The most important thing in this regard is to understand what is inside the theory and what is outside its scope - where, it turns out, a lot of the striking peculiarities hang out.<br />
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First, let me say that there are <i>different interpretations </i>of QM. An interpretation is the way one writes down the theory, starting from the assumptions and going all the way to predictions of experiments that we can carry out to test the theory. So the different interpretations should <b>not </b>be thought of as different theories because they all end up predicting the same results (the results of QM experiments), but they differ - sometimes significantly - on the philosophical level. The standard interpretation of QM is called the <i><a href="http://simplyphy.blogspot.ch/2015/01/a-trip-to-copenhagen.html">Copenhagen interpretation</a> </i>because of the place where its basis was developed, by scientists like Niels Bohr and Werner Heisenberg. This is the interpretation that is taught in school and university, and in fact it is so well-established that I was already doing my Master's degree when I realized it is not the only one. Thus, the rest of this discussion is in view of the Copenhagen guys' way of seeing the quantum world.<br />
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So, what goes in a physical theory? The first thing is the <i>ontology </i>of the theory, which according to <a href="http://en.wikipedia.org/wiki/Ontology">wikipedia</a>, '...deals with questions concerning what entities exist or can be said to exist...' In simpler terms the ontology is what your theory is about, what the object of your study is. For example, Newton's theory of gravity describes how massive objects attract each other, thus the ontology of the theory is 'massive objects'. The stuff in the ontology has the right of existing, of <b>be</b>ing in an absolute, a priori sense. Once we have the ontology, the rest of the theory is comprised by the laws that govern the system, and, importantly, that tell us how to <b>observe </b>stuff, because we would like our theory to be telling us something about the real world. Now, in Newton's gravity, <b>be</b>ing and being <b>observed </b>are pretty much the same thing: the laws determine where a massive particle <b>is</b>, and if we look for it, that's also where we are going to <b>observe </b>it. But it turns out that this is not the case in QM, and understanding the distinction between the objective <b>be</b>ing, and the subjective <b>observation </b>of being is the key to resolving some of the peculiarities.<br />
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The ontology of QM is the wave function. There is nothing else in there. The rest of the theory (within the Copenhagen interpretation) is the Schroedinger equation, which tells us how to compute the wave function of a system, and the measurement postulates, which tell us how to compute the outcome of an observation. So, very generally, there are two types of questions we can ask: 'what <b>is</b> the wave function?' and 'what do we expect to <b>observe</b> if we do such and such a measurement?'. Any other question would be outside the scope of the theory and so meaningless to ask - a bit like trying to divide by zero, or asking Newton's gravity how a charged object would move in an electric field.<br />
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If that doesn't sound at all simple - or important, maybe some practical examples will help. I'll cover some of the most common QM conundrums, always starting with a statement which one could sometimes hear, but which is wrong, then explaining why.<br />
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<i><a href="http://simplyphy.blogspot.ch/2015/04/down-with-duality.html">Wave-particle duality</a></i><br />
<u>Wrong statement</u>: A quantum object is simultaneously a wave and a particle.<br />
<u>In fact</u>: Remember our ontology: it is exclusively the wave-function. So the object <b>is </b>a wave (function). The particle nature comes second: it is only needed to explain outcomes of <b>observations</b>, like the fact that if we try to observe electrons, they appear at a very well-defined position, as if they're point-like particles hitting our detector. So quantum objects are fundamentally waves; we are only used to thinking of them as particles in some cases because of the way we are used to observing them.<br />
<u>Correct statement</u>: A quantum object <b>is</b> a wave function, which in <b>observations</b> sometimes appears as a wave, but sometimes shows properties we associate with particles.<br />
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<i>Superposition states</i><br />
<u>Wrong statement</u>: In QM, a particle can be in two places at the same time.<br />
<u>In fact</u>: There is no 'particle' in our ontology. There is only the wave function, which is the only thing that can <b>be </b>such and such. The question 'what <b>is</b> the position of the particle' is not among the questions that can be answered by the theory, since the only questions about <b>be</b>ing we can ask are about the wave function. Now, it is true that there <b>is </b>a wave function which corresponds to a particle being <b>observed </b>at a given position with a probability 1. It is also true that we can take such a function, which places the particle at position A, and another one which places the particle at a different position B, sum them up, and get a valid wave function which would correspond to <b>observing </b>the particle at either A or B with probability 1/2 each. But we will always observe the particle in exactly one of the two positions. Before that, the particle is not at two places at the same time. The only thing that <b>is </b>is the wave function, which is what it is, duh. One of my biggest pet peeves is when <a href="https://www.ted.com/talks/aaron_o_connell_making_sense_of_a_visible_quantum_object?language=en">scientists who should know better make wrong statements</a> regarding superposition, which are always hugely misleading for the public.<br />
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<i>Uncertainty principle</i><br />
<u>Wrong statement</u>: In QM, a particle cannot <b>be</b> in a well-defined position with a well-defined velocity.<br />
<u>Correct statement</u>: We cannot simultaneously <b>observe</b> the exact position and the exact velocity of a quantum object.<br />
The key is, as usual, in what is allowed to <b>be</b>.<br />
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<i>Determinism</i><br />
<u>Wrong statement</u>: QM states that the world is intrinsically non-deterministic.<br />
<u>In fact</u>: The wave function evolves in a deterministic way, as given by the Schroedinger equation. The outcomes of measurements are, however, indeed probabilistic. So the universe - the wave function - follows well-defined laws, but our observation of it has some in-built randomness. It is by the way very commonly said that Einstein could not accept this and that he famously said, 'I don't believe God plays dice'. This is, again, not exactly true. Even if Einstein had some difficulty accepting probabilistic outcomes of measurements, at some point he was willing to live with it. His much more important objection against QM was its apparent <i>non-locality</i>, which is a vast topic and to a large extent still an open question (but I won't discuss it in this post).<br />
<u>Correct statement</u>: In QM, what <b>is</b> is deterministic, what is <b>observed</b> is probabilistic.<br />
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By the way, Einstein's full quote actually goes, "I don't believe God plays dice since the intervention."<br />
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<i><a href="http://simplyphy.blogspot.ch/2015/03/jigsaws-cat.html">Schroedinger's cat</a></i><br />
<u>Wrong statement</u>: The cat in Schroedinger's thought experiment is simultaneously dead and alive.<br />
<u>In fact</u>: The cat will be <b>observed </b>either dead or alive, with probability 1/2. Outside of that the cat is not being a cat, but instead <b>is</b> a quantum object (composed of a huge number of quantum objects, but that doesn't matter) which <b>is </b>a wave function in a superposition state, see above.<br />
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Bottom line: everything in the quantum universe <b>is </b>a wave function, and nothing else can <b>be</b> anything. Once we accept this, it is not the reality which is strange, it is only the outcomes of measurements which appear so. I realize all this might not be a very satisfying explanation. For example, it's hard to accept that the cat is in fact not a cat but a wave function, and we're only observing it (in the most literal sense: by looking at it) to be a cat. But I do think that if everybody kept in mind the important distinction between the physical reality and the results of its observation, and hence between valid statements and those which are wrong simply because they are meaningless within the scope of the theory, the world would be a better place.<br />
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By the way, all this is not, like, 'just my opinion'. I mean, I cannot go as far as to say that everything written here is correct and whoever disagrees is an idiot, but I guarantee that the 'wrong statements' above <b>are</b> wrong, and I'm willing to argue to the death with anybody about this. Also, Bohr himself supposedly often emphasized that the answer to most 'paradoxes' of QM lies in the fact that people are asking questions they are not allowed to ask. So, you know... I like to think that even he would agree with this post, if he could see it.<br />
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Perhaps you don't see an important difference between the correct and the wrong statements above. Perhaps you do, but it seems to you that I substituted one 'strangeness' with another. Or everything is clear? Whichever the case, all this is just scratching the surface. Everything that is in italics within this post deserves a post on its own, and will get one in due time.<br />
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So... stay tuned!<br />
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<i>Edit: It's become clear to me that I shouldn't be talking so freely about 'ontology'. Here, by 'ontology' I mean whatever the theory is about, the 'reality' of the theory. There is a deeper philosophical question which is whether what the theory is about is also what constitutes the reality we live in, or whether reality is something else that just behaves in a way that's predicted by what the theory is about. The two points of view are called 'ontological' and 'epistemological', respectively, where now 'ontology' refers to the actual, physical reality. This debate is still widely open when it comes to the Copenhagen interpretation. Honestly, to me this question is a dead end: I can't even conceive of a way one could begin to resolve it. Thus, it was never my intention to go into these purely philosophical matters - in fact, I wanted to keep philosophy to the minimum and only talk about what is found within the well-defined framework of the 'standard' Quantum Mechanics. The main goal of the text, to summarize it in one sentence, is to explain that there is no objective 'being' in QM apart from the being of the wave function and the being of results of measurements. </i></div>
Kozmohttp://www.blogger.com/profile/17418168704940582871noreply@blogger.com0