Distinguishing identical particles

The problem is within the description of the system after your first measurement. You say particle A is in state $\psi_{1}$ and particle B is in state $\psi_{2}$. This state could be written down as

$$\psi_{1}(x_A)\psi_{2}(x_B)$$

However, given that the two particles are identical, this is not a valid quantum state. The reason this state is not valid is not straightforward, but it boils down to the fact that you would get different predictions for some observables depending on the labels you chose for the particles. Since the particles are identical this would not be physically possible. The only valid quantum states are of the form

$$\psi_{1}(x_A)\psi_{2}(x_B) \pm \psi_{2}(x_A)\psi_{1}(x_B)$$

The problem with your reasoning is that you assumed that your first measurement allowed you to distinguish the particles and assign a label to them. That is in fact not possible, and there lies the loophole in your argument.

Let me try to understand what you are proposing: If there are two different stationary states, then they must have different energies, unless they are degenerate states. You stated that the two identical particles are non-interacting, thus they cannot exchange energy. This would mean that each particle would be stuck in their stationary states and we'll be able to distinguish them apart.

So I see that there's no other energy involved, both particles have different energies and therefore will be in different stationary states. I think the problem is that you state they are non-interacting. The two particles are distinguishable by construction.

At first I thought you were talking about position measurements, if so read the following:

You can see the problem clearly when their probability density overlaps in the same region of space. Notice that if you draw out the stationary state of both particles (no matter which states you choose), you have to overlap them because both of these are in the same well! There will be overlap for certain places. So, if you find out that one particle's position is where the overlap is, you won't know if that came from the first stationary state or the second.

Try this example: Take an O2 molecule. How can you tell which electron belongs to which nuclei?

(In)distinguishability is always relative to a chosen level of description.

Once you label particles by the eigenstate they occupy they become distinguishable. This means that for their description you changed the assumptions about the accessible state space, restricting it to eigenstates (by not allowing external forces that would perturb the Hamiltonian).

This happens generically for electrons in quantum chemistry, where they are classified as to which shell they belong to, based on spectral information in the Hartree-Fock approximation. These labelled electrons behave like distinguishable particles, as the effective state space has been reduced.

See also the entry ''Indistinguishable particles and entanglement'' in Chapter B3: Basics on quantum fields of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html