How do we know that the two indistinguishable particles in the same infinite well have different energies?

Question

I'm reading an example in which we have two identical particles in the same infinite well. They have different quantum numbers "n", which means that they have different energies. This example is used to introduce us to the need for the symmetry requirement.

I'm confused as to how we know the particles have different energies if we can't tell them apart? The way I see it, if we're not able to identify a location in space in which one particle can be found but the other one isn't, or make any other observation that would distinguish the two particles, how do we know they have different quantum numbers and thus different energies? It seems that for all we know they have the same energy and everything about them is the same.

Answer 1

If I have two balls in a box, with energies $E_1$ and $E_2$, and they're painted different colors, I can say "the blue one is the one with energy $E_1$, and the red one is the one with energy $E_2$." That's what it means to be distinguishable. When you have identical particles, you can't do that, you can only say that one of them has energy $E_1$ and one of them has energy $E_2$.

Suppose you close the box and then open it again later, and at this point there are still balls with energies $E_1$ and $E_2$. Is the ball with energy $E_1$ the same one as the ball with energy $E_1$ earlier? If the balls are distinguishable, you can tell by the color. For identical particles, it's a meaningless question. (And this isn't just some argument over semantics: it affects the experimental predictions, because it changes how we do the counting in calculations.)

Answer 2

Saying that there are two particles in the system where one has energy $E_1$ and the other has energy $E_2$ doesn't contradict the particles being indistinguishable. This is because we haven't labeled the particles in a way like "particle $A$ is the particle with energy $E_1$ and particle $B$ is the particle energy $E_2$".

This is why, say for electrons, we would then know the system is in the anti-symmetric state $$|\psi\rangle=|E_1,E_2\rangle-|E_2,E_1\rangle$$

Answer 3

The other answer is entirely correct; however, I think this confusion is more common than we realize, and it's a result of language colloquialism.

For instance, when we learn about the Periodic Table and the shell model, say we're at Helium, we're told, "and then we add another electron, but the $1S$-shell is filled so the electron goes in to the 2S shell, making lithium's atomic structure [He]$2S^1$".

Well that's not the case really. You add the electron and there is no way to antisymmetrize 3 particles in the two $1S$ states, so you bring in $2S$ and the 3 electrons go into a antisymmetric combination in which no particle label has a definite energy.

It's quite different from what we're told: the 2 electrons in the ground states saying, "no room here", and the third one says, "OK, I'll stay up here in next available state".

Answer 4

The energies of a particle in the infinite well are given by $$E_n = E_1 n^2.$$ For two particles we have $$E = kE_1 = E_n + E_m = E_1(n^2 + m^2).$$ there are no many ways how an integer can be decomposed into a sum of two squares of integers, $k = n^2 + m^2$. Thus by measuring the total energy it is possible to say, whether the two particles are in different states or not.

Life would be harder, if instead of the infinite well there was a harmonic oscillator ;)

Answer 5

There are already excellent answers here but let me just reiterate in a slightly different manner.

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When we say two indistinguishable particles have energy $E_1$ and $E_2$ we don’t know which particle has which energy. What we’re saying is that when we measure the energy of a particle, the probability of getting $E_1$ or $E_2$ is equal. Moreover, the probability of getting any other energy is zero.

Indistinguishability just means that a priori we can’t say which one of the two energies will be the outcome.