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Rutherford did say something like this but suggesting a barmaid instead of a six-year-old, while Hilbert suggested the first man on the street. Technically, if we assume the statement correct, then nobody has understood anything about science, ever. But I'd like to have my go at explaining Physics-y stuff as simply as I can, and, perhaps equally importantly, without too much of the "woaaah quantum mechanics is so weird, you just have to accept it like this even though it's completely non-intuitive" crap one often finds in science articles written for the laymen.

Friday, January 30, 2015

A trip to Copenhagen

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.

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 principles, you'll notice some differences. Anyway. I do not claim any formal rigor here, in fact I am trying to put the essential postulates in words as simple as I can find. 
  1. 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.
  2. 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.
  3. A measurement apparatus has an in-built set of mutually exclusive (see postulate 5) states Ψ1,  Ψ2, ..., 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.
  4. 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. Ψ2 if the dial shows '2'. It then proceeds to evolve smoothly in time until the moment in which another measurement is made.
  5. 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 Ψn. 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 Ψi, it has a probability of 0 to be measured in any of the other states Ψn, where j is not i.
  6. (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). 
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. 

Here are your two geeky jokes:
1. Schroedinger's cat walks into a bar. And doesn't.
2. A Buddhist monk at a hod-dog stand says, 'make me one with everything.'

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 this. In fact I wanted to put two jokes about quantum mechanics here, but I skimmed through some websites and did not find a single one which was both original and did not make me want to be like, 'well, that's not exactly right!' So I decided to go with the Buddhist one, which always makes me chuckle. :)

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