The occupation-number representation

Suppose each of our particles can sit in any one of $M$ states $\{\vert\phi_i\rangle \}$. These could be, for example

• Orbitals with a particular spin localized on or near a particular atom (in solid-state physics); or
• Positions in a discretized version of real space.

Let's assume for convenience these states are orthogonal (though that is not essential, so long as they are linearly independent). Then we can construct a complete basis set for the $N$-particle system from the symmetrized products:

$$\vert i_1,\ldots,i_N\rangle = {1\over\sqrt{N!}}\det\left\vert\begin{array}{ccc} \phi_{i_1}(r_1)... ..._{i_N}(r_1)&\ldots&\phi_{i_N}(r_N) \end{array}\right\vert\quad\hbox{(fermions)}$$ (55)

$$= {1\over\sqrt{N!}}{\rm perm}\left\vert\begin{array}{ccc} \phi_{i_1... ...i_{i_N}(r_1)&\ldots&\phi_{i_N}(r_N) \end{array}\right\vert\quad\hbox{(bosons)}.$$ (56)

We can alternatively specify each $\vert i_1,\ldots,i_N\rangle$ by the occupation number $n_j$ of each state $\phi_j$: this is simply the number of times state $j$ sppears in the set $\{i_1,\ldots i_N\}$. For fermions the occupation number must be 0 or 1; for bosons it can be any integer. Let us divide the single-particle basis set into two subsets, corresponding to the system $S$ and the environment $E$. Now the problem has a direct product structure: to specify a configuration of the system (one of the basis kets $\{\vert i_1,\ldots,i_N\rangle \}$) we have to specify the values of the occupation numbers for both the system region and the environment region. If we know there are exactly $N$ particles in the system, there is a constraint on the occupation numbers that

$$\sum_{i\in S}n_i+\sum_{j\in E}n_j=N.$$ (57)

However, if the system is very large then it generally makes no difference whether we work at fixed $N$ or at fixed chemical potential; in that case, no constraint on the occupation numbers is necessary.