The Hamiltonian for an ideal gas of N particles is
The eigenvalue problem for the Hamiltonian is in the form of the time-independent Schrödinger equation for the (unsymmetrized) eigenfunctions
First, we notice that the equation is completely separable in the N
particle coordinate/spin labels 
, meaning that the Hamiltonian
is of the form
Note, further, that H is independent of spin, hence, the eigenfunctions
must also be eigenfunctions of 
and 
.
Therefore, the solution
can be written as
where 
is a single particle wave function
characterized by a set of spatial quantum numbers 
and eigenvalues 
. The spatial quantum numbers
are chose to characterized the spatial part of the
eigenfunctions in terms of appropriately chosen observables
that commute with the Hamiltonian.
Note that each single-particle
function can be further
decomposed into a product of a spatial function 
and a spin eigenfunction 
, where
Substituting this ansatz in to the wave equation yields a single-particle wave equation for each single particle function:
Here, 
is a single particle eigenvalue, and
the N-particle eigenvalue is, therefore, given by
We will solve the single-particle wave equation in a cubic box of side L for single particle wave functions that satisfy periodic boundary conditions:
Note that the momentum operator 
commutes with the corresponding
single-particle Hamiltonian
This means that the the momentum eigenvalue 
is a good number
for characterizing the single particle states 
. In fact, the solutions of the single-particle
wave equation are of the form
provided that the single particle eigenvalues are given by
The constant C is an overall normalization constant on the
single-particle states to ensure that 
.
Now, we apply the periodic boundary condition. Consider the boundary condition in the x-direction. The condition
leads to
or
which will be satisfied if
where 
is an integer, 
,.... Thus, the momentum 
can take on only discrete values, i.e., it is quantized, and
is given by
Applying the boundary conditions in y and z leads to the conditions
Thus, the momentum vector can be written generally as
where 
is a vector of integers 
.
This vector of integers can be used in place of to characterize
the single-particle eigenvalues and wave functions. The single-particle
energy eigenvalues will be given by
and the single-particle eigenfunctions are given by
Finally, the normalization constant C is determined by the condition
Therefore, the complete solution for the single-particle eigenvalues and eigenfunctions is
and the total energy eigenvalues are given by
Another way to formulate the solution of the eigenvalue problem is
to consider the single particle eigenvalue and eigenfunction
for a given vector of integers 
:
and ask how many particles in the N-particle system occupy this
state. Let this number be 
. is called an
occupation number and it tells just how many particles occupy the
state characterized by a vector of integers . Since there are an
infinite number of possible choices for , there is an infinite
number of occupation numbers. However, they must satisfy the obvious
restriction
where
and
runs over the (2s+1) possible values of m for a spin-s particle. These occupation numbers can be used to characterize the total energy eigenvalues of the system. The total energy eigenvalue will be given by
