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I am reading chapter 6 of Sakurai's Modern Quantum Mechanics and have come across the 'symmetrization postulate', which tells me that for any given system of identical particles, all states must either be symmetric with respect to the exchange operator, or antisymmetric.

Why can we not have some states that are symmetric with respect to the exchange operator, and some that are antisymmetric? As far as I can tell, this question is not even referenced anywhere in the material I've been reading.

I understand that this is the symmetrization postulate, but is there any chance anyone could elucidate on why this is the case? Is this a consequence of quantum field theory?

Qmechanic
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king_geedorah
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1 Answers1

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If particles are identical they follow exactly the same rules, therefore any operator on any of them affect any other particle in the same exact way. You can not choose or find any operator that would act on some in one way and on some others in a different way since they are identical. Symmetry or anti-symmetry comes when you think what could happen when to exchange two particles (not that you move them around, that would be a current, but rather you just swap with the other): $$\psi(r_1, r_2) \rightarrow \psi(r_2, r_1)$$ and you can always extend this to any number of particles since reorganizing any number of them can be done by pairs. In that case the state of the system should maintain $$|\psi(r_1, r_2)|^2$$ unchanged since they are identical. Then, under any reorganization of any number of identical particles any exchange of two of them would result in either symmetry or anti-symmetry of the wave-function or state.

Nelson Vanegas A.
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