quantum mechanics - Distinguishing identical particles

I've been going through Shankar's Principles of Quantum Mechanics. In the section of the system of identical particles, he uses an example of billiards to illustrate the difference between identical particles in classical vs. quantum mechanics.

He argues that in classical mechanics, we can track the history of a particle (a billiard ball) to distinguish it from another particle with no intrinsic differences. In quantum mechanics, however, he argues since continual observation is not possible, we can't use the same method to distinguish identical particles.

A potential counter-example I thought was...suppose we have two non-interacting particles in a same square well. And at the end of some measurement, we find that one particle 1 is in a stationary state $\psi_1$, and particle 2 in $\psi_2$. We measure the system again after some time t, then we know whichever particle that's in $\psi_1$ must be particle 1 from the previous measurement. And the same goes for particle 2. Thus we can distinguish the two "identical" particles.

What conceptual mistakes am I making here?

Answer

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?