Schrodinger equation for N particles in a box

[Kashmir]

[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

 

[anuttarasammyak]

We should investigate whether N particle of same kind are bosons or fermions for their placement in energy states.

 

[Kashmir]

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.

 

[Kashmir]

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?