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A friend asked me today about a question that appeared in the Spanish national examination for being a radiophysicist (in hospitals). He thinks this question is wrong.

It says the following:

In quantum systems of many identical particles, point out the wrong sentence:

  1. The fact of the particles being identical requires that the Hamiltonian of the system is symmetrical w.r.t. the variables that describe the system.
  2. For an applied perturbation to a quantum system, it is not mandatory that the condition of identity of the particles is preserved.
  3. A completely symmetrical state of the system will keep being completely symmetrical at any time.
  4. Symmetry and antisymmetry properties of wavefunctions are not due to attraction or repulsion but to the indistinguishability of particles.

I am sure 4 is true since symmetry or antisymmetry are related to particles being indistinguishable bosons or fermions (even though 4 is poorly phrased from my p.o.v.).

When going to 3, I guess they mean that, if you have a set of symmetry operators $\Lambda_i$, then a completely symmetrical state would be the one that is an eigenstate of all the symmetry operators $\Lambda_i |\psi\rangle = \lambda^j_i |\psi \rangle$, being $\lambda^j_i$ the $j$-th eigenvalue for the $i$-th symmetry operator. Since a symmetry operator commutes with the Hamiltonian, it will also commute with the evolution operator and thus an eigenstate of all the symmetry operators will keep being symmetrical in time.

About 1 (also very poorly phrased I'd say), I don't know sincerely. You could phrase it the other way around. 'If the Hamiltonian is not symmetrical wrt the variables, then the particles are not identical'. In this phrasing you clearly see that this is false. However, it occurs that in Spanish they used a phrasing that can be understood as symmetric Hamiltonian implies identical particles or exactly the other way around, so I would mark this as false if I were sure the others are true.

About 2, I don't know what they mean at all. Are they trying to speak about Zeeman/Stark splitting? Like if you have a system with identical particles for a hamiltonian but you introduce a new term, then it may be that you can distinguish them? I don't know.

If anyone can help, I'd be really grateful : )

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    $\begingroup$ In Russia radiophysics means physics of radio waves, etc. I am quite amused by its different meaning in Spain (though I see the logic). Sorry for this irrelevant comment. $\endgroup$ Feb 2 at 9:02

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I think the answer is 2.

1 - True. I think the meaning is that the Hamiltonian must be invariant to permutation of the identical particles (i.e. relabeling the particles).

2 - False. A perturbation that can distinguish between particles is necessarily an interaction that makes these particles not identical and thus not possible. This is somewhat a semantic issue, but I think experimentally this is also a solid statement since applying an interaction to a specific particle out of many identical ones is impossible.

3 - True. I think the meaning is exactly what you are referring to, the symmetry operator commutes with the Hamiltonian and the time evolution operator, thus a symmetric state retains its parity.

4 - True. This is exactly bosonic-fermionic statistics and all their following properties.

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