{"id":4525,"date":"2016-01-01T17:25:18","date_gmt":"2016-01-02T00:25:18","guid":{"rendered":"http:\/\/www.wall.org\/~aron\/blog\/?p=4525"},"modified":"2018-09-05T13:07:14","modified_gmt":"2018-09-05T20:07:14","slug":"quantum-mechanics-i-interference","status":"publish","type":"post","link":"https:\/\/www.wall.org\/~aron\/blog\/quantum-mechanics-i-interference\/","title":{"rendered":"Quantum Mechanics I: Interference"},"content":{"rendered":"

A bunch of people have been asking me about the interpretation of QM.\u00a0 Now, every interpretation of QM predicts (or claims to predict) the same experimental results in any experiment (or at least, any realistically feasible experiment).\u00a0 Otherwise they wouldn’t be rival interpretations<\/em>, they would be rival theories<\/em>, and we would just do an experiment to see who is right.<\/p>\n

So before discussing what QM actually means, it’s good to get the ground rules down\u2014the ones that all physicists agree are the right ones to use to predict the results of actual experiments.<\/p>\n

Let’s suppose we’re doing a physics experiment, which I am going to describe in an extremely abstract way, because that’s the kind of person I am.\u00a0 A (somewhat idealized) way to describe a certain class of experiments is as follows:\u00a0 We start out by preparing the initial configuration of the apparatus to be in some particular configuration (or “state”), let’s call it A<\/strong>.\u00a0 For example, we start with a radioactive atom.\u00a0 We can let the experimental apparatus evolve on its own, isolated from the rest of the world, until it reaches some final configuration.\u00a0 Then we look inside, e.g. 10 minutes later, and check what the current state of the experiment is.\u00a0 Perhaps the atom has now decayed into something else.\u00a0 Let’s call this final configuration B<\/strong>.<\/p>\n

Several aspects of this description are clearly idealizations.\u00a0 There are always some limitations in our ability to control and\/or know the initial condition A<\/strong>; the system is never going to be completely isolated from the rest of the world no matter how hard I try.\u00a0 And in some experimental setups this may be a good thing\u2014that is, we may want to deliberately reach in to measure and\/or adjust the system, part way through its “time evolution”.\u00a0 (Unlike biologists, we physicists use the word evolution<\/em> any time anything changes!) \u00a0 And, at the end of the process, we’re never going to be able to measure the final outcome with perfect precision either.<\/p>\n

But I’m a theorist so I can ignore the messiness of real life, whenever it pleases me to do so.<\/p>\n

Now if the laws of physics were deterministic (and if we know what they are, and we know the initial state completely precisely…) then in principle<\/em> we could simply solve all of the relevant equations and find out what exactly the final state would be.\u00a0\u00a0 So after 10 minutes, any given A<\/strong> will become some particular B<\/strong> with probability 1.<\/p>\n

(In practice, this calculation is often impossible because of phenomena like chaos<\/em> where (in some systems, not others) the final outcome depends very, very sensitively on the initial conditions.\u00a0 For chaotic systems, you need to know the initial conditions to exponential precision<\/em> in order to predict the future.\u00a0\u00a0 This is why we can’t predict the weather accurately for more than about a few days out, because the number of digits accuracy you’d need to measure things to is proportional to the number of days!)<\/p>\n

But the actual laws of physics are stranger than that.\u00a0 They are not<\/em> deterministic.\u00a0 I think I’ve read that some Philosophers of Science claimed that Determinism was an important assumption underlying the possibility of doing Science at all.\u00a0 Well, Determinism is false, and yet we scientists still have jobs.\u00a0 I know, 20-20 hindsight, but it was still a dumb thing to say (if anyone in fact ever said it, which I haven’t done the research to confirm…).<\/p>\n

So let’s try again.\u00a0 Once again, we’ll set up our experiment in a particular initial state A<\/strong>.\u00a0 But now, there are several possible final states B<\/strong>, B’<\/strong>, B”<\/strong> etc.\u00a0 Let’s suppose we want to calculate the probability for some specific one: B<\/strong>.\u00a0 So the sane, sensible way of doing this, would be to think of all the different ways that A<\/strong> could evolve in time to become B<\/strong>.\u00a0 To actually do calculations, you apply the rules of probability theory:<\/p>\n

NORMAL PROBABILITY THEORY:<\/p>\n