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*single-particle state*.

By this, is it meant that if the particles can be assumed independent, then each particle can be described by an individual wave function, with the entire system being described by a wave function that is a product of these

*single-particle*wave functions?

If this is the case, is it then true that since each particle has an independent Hamiltonian associated with it (since the particles are independent, there are no interaction

*cross-terms*in the potential), and hence the energy of the system is the sum of the energies of the energies associated with each single-particle wave-function?

If the particles are identical and indistinguishable is it correct to say that one can no longer distinguish which particle is in which state precisely and instead one has to quantify things in terms of the number of particles occupying each single-particle state, and the particular state is identified in by its corresponding energy. One then describes the microstate of the entire system in terms of the occupation numbers of each single-particle state with a given energy.