I Schrodinger equation for N particles in a box

Kashmir
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[Pathria, statistical mechanics][1], pg2 ,when discussing ##N## particles in a volume ##V##

"...there will be a large number of different
ways in which the total energy E of the system can be distributed among the N particles
constituting it. Each of these (different) ways specifies a microstate, or complexion, of the
given system"
"Each of these (different) ways specifies a microstate, or complexion, of the
given system. In general, the various microstates, or complexions, of a given system can
be identified with the independent solutions ##ψ(r_1,..., r_N )## of the Schrodinger equation of
the system, corresponding to the eigenvalue E of the relevant Hamiltonian" .
If we have N particles in a box, we solve the energy eigenvalue for the total system ##H|E>=E|E>## and find the eigenvectors.

Using linear combination of these and boundary conditions we find the general solution ##ψ(r_1,..., r_N )##.So how does "the various microstates, or complexions, of a given system can
be identified with the independent solutions ##ψ(r1,..., rN )## of the Schrodinger equation of
the system, corresponding to the eigenvalue E of the relevant Hamiltonian" enter the discussion? [1]: https://www.google.co.uk/books/edition/Statistical_Mechanics/PIk9sF9j2oUC?hl=en
 
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We should investigate whether N particle of same kind are bosons or fermions for their placement in energy states.
 
anuttarasammyak said:
We should investigate whether N particle of same kind are bosons or fermions for their placement in energy states.
The author didn't mention that.
 
Here is my attempt:
Suppose we have two non interacting particles in identical environment.

The individual Hamiltonian for both is ##H_1## and the total Hamiltonian is ##H=H_1 \otimes 1+ 1\otimes H_2##
Also if ##|\phi_1>## and ##|\phi_2>## are the eigenstates of ##H_1## with eigenvalues ##E_1##,## E_2## then the eigenstates of the system is ##|\phi_1> \otimes|\phi_2>## with eigenvalue ##E_1 + E_2##

One of the state of the system is then
##|\psi >= e^(-iE_1t/\hbar)|\phi_1>
\otimes e^(-iE_2t/\hbar) |\phi_2>##

This state corresponds to particle 1 in state ##|\phi_1>## having energy ##E_1## & particle 2 in ##|\phi_2>## with energy ##E_2## defining a microstate.

Is this correct?
 
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