5.2 Ideal quantum gas: Grand canonical ensemble

We may derive the properties of a quantum gas in another way, making use of the ($z,V,T$) ensemble in Gibbsean phase space. Recalling the general definition of the grand partition function, $Z(z,V,T) \equiv \sum_{N=0}^{\infty} z^{N} Q(N,V,T)$, we now write $Q$ as a sum (in place of an integral) over states:

$$Q(N,V,T) = \frac{m^{3N}}{N! h^{3N}} \int d\vec{\Gamma} \exp ... ...\exp \left[ -\sum_{\vec{n}} f_{\vec{n}} E_{\vec{n}}/kT \right]$$ (5.5)

The sum $\left\{ f_{\vec{n}} \right\}$ is to be taken over all permitted population numbers of all states $\vec{n}$, again requiring that $\; \sum_{\vec{n}} f_{\vec{n}} = N$. The permitted values of $f_{\vec{n}}$ are: $0$ and $1$ for fermions, and $0,1,2, \dots N$ for bosons. In this manner we arrive at

$$Z = \sum_{N=0}^{\infty} z^{N} \sum_{\left\{ {\large f}_{\vec... ...{\vec{n}} \left( z e^{-E_{\vec{n}} / kT} \right)^{f_{\vec{n}}}$$ (5.6)

It is easy to show that this is equal to

$$Z = \prod_{\vec{n}} \left[ \sum_{f} \left( z e^{-E_{\vec{n}}}\right)^{f} \right]$$ (5.7)

Now we can insert the possible values of $f$. We find that for

Fermions ($f=0,1$):

$$Z=\prod_{\vec{n}} \left[ 1+ze^{-E_{\vec{n}}/kT}\right]$$ (5.8)

and for Bosons ( $f=0,1,2, \dots$):

$$Z=\prod_{\vec{n}} \frac{1}{ 1-ze^{-E_{\vec{n}}/kT} }$$ (5.9)

Having secured the grand partition function, we can now apply the well-known formulae for pressure, mean particle number, and internal energy (see Section 4.3) to determine the thermodynamic properties of the system. The mean population number of a given state $\vec{n}$ is

$$\langle f_{\vec{n}} \rangle \equiv \frac{1}{Z} \sum_{N} \sum... ... - \frac{1}{\beta} \frac{\partial}{\partial E_{\vec{n}}} \ln Z$$ (5.10)

Inserting for $Z$ the respective expression for Fermi or Bose particles we once more arrive at the population densities of equ. 5.4.

In the following sections we discuss the properties of a few particularly prominent fermion and boson gases.