Quote:
Originally Posted by Max Raker
I probably put much more physics terminology into my question than was needed, sorry. In quantum mechanics, if you have a box full of electrons, you actually can't tell the electrons apart from one another and this turns out to greatly limit the type of mathematical functions we can use to describe their behavior. Classically, it is possible to sort of put labels 1 to N on each electron in the box, but the uncertainty principle does not allow this and this is actually a very important distinction in terms of physical properties.
I'm sure it doesn't impact your problem, but it is a very interesting and physically deep application of statistics to physics, based on some very basic math to somebody with your background.
This is interesting and sounds vaguely familiar, but I think I need help understanding. Please forgive and correct any stupid mistakes I make below. It has been a long time since I thought about the details of QM, and I will try to discuss this from memory.
Suppose we have two particles in a one-dimensional box. In this case, the wave function is some f(x,y) ∈ L
2(
R2). At this stage of the modeling, the particles are labeled, at least in the sense that the "first" particle is associated with the x-variable, and the "second" with the y-variable.
Now, if we measure the positions of the particles, the probability that the first particle is in [a,b] and the second is in [c,d] is
This need not be the same as the probability that the first particle is in [c,d] and the second is in [a,b]. So even at this level, the model is making a labeled distinction between the two particles.
Are you simply saying that if we take a snapshot of the two particles, then it is impossible to tell, with certainty, which of the two particles in the snapshot corresponds to the x-variable in the wave function? Or, to put it another way, if we take two consecutive snapshots, it is impossible to tell, with certainty, which particle in the second snapshot corresponds to which particle in the first.
I am probably misunderstanding what you are saying, but if this is the assertion, then why is this not also a problem in classical models of statistical mechanics? If two heavy particles are performing Brownian motion in an ambient medium and we take two consecutive snapshots, then mathematically (according to the probabilistic model) it is not possible to determine, with certainty, which particle in the second snapshot corresponds to which particle in the first.