I've been reading chapter 10.3 'Identical Particles' in Shankar's book on quantum mechanics and also looked through some of other books on this subject and one rather subtle objection started bothering me.
They all argue that unlike classical physics, in which you can follow the trajectories of the particles without disturbing the states, there exists no physical basis for distinguishing between identical particles in quantum mechanics where any measurement leads to collapse of the state. And they straight go to saying that this in turn implies that there should be restriction imposed on the system of identical particles that two configurations related by the exchange of identical particles must be treated as one (=invariant under exchange operator) and be indistinguishable.
But as far as I understood, to be Identical only means that all the Internal Tags such as spin, mass, charge etc., which allows the observer to distinguish between particles without referring to their positions or momentums, are all the same. And here's what's been bugging me. I thought that whereas it is necessary to be 'Identical' in order for the system to be 'Indistinguishable' under the exchange of the particles, particles being 'Identical' is not sufficient for the particles to be 'Indistinguishable' .
Say, there existed a state corresponding to two identical particles and that is some nontrivial superposition of symmetric and antisymmetric states, i.e.,
then it is clear upon acting the exchange operator that this state is a 'Distinguishable' yet being one possible state of two identical particle system.
So it is clear from this that for the system to be indistinguishable, it requires them to follow either "Fermi-Dirac statistics" or "Bose-Einstein statistics" (either completely symmetric or completely antisymmetric) in addition to them being identical in their intrinsic properties such as spin, mass, charge, etc., and that Indistinguishability and Identicalness better be kept separate. I do agree that it can be postulated that all systems of identical particles must choose to be totally symmetric or antisymmetric (thereby becoming indistinguishable) and verify this postulate by experiments after experiments, but this indistinguishability should not be taken as something naturally derived from identicalness of the system.
Please verify if I'm right here or convince me with the legitimate way to address this issue if I'm wrong.
I edit my question cause I found myself with a video from MIT opencourseware that supports the idea that it should be made an extra postulate.
It starts on 1:10
Check this out!