1549380323-Statistical Mechanics Theory and Molecular Simulation

11

The quantum ideal gases:

Fermi–Dirac and Bose–Einstein

statistics

11.1 Complexity without interactions

In Chapters 3 through 6, the classical ideal gas was used to illustrate how the tools of
classical statistical mechanics are applied to a simple problem. The classical ideal gas
was seen to be a relatively trivial system with an uninteresting phasediagram. The
situation with the quantum ideal gas is dramatically different.
The symmetry conditions imposed on the wave function for a systemofNnon-
interacting bosons or fermions lead to surprisingly rich behavior. For bosonic systems,
the ideal gas admits a fascinating effect known asBose–Einstein condensation. From
the fermionic ideal gas, we arrive at the notion of aFermi surface. Moreover, many of
the results derived for an ideal gas of fermions have been used to develop approxima-
tions to the electronic structure theory known asdensity functional theory(Hohenberg
and Kohn, 1964; Kohn and Sham, 1965). Thus, a detailed treatment of the quantum
ideal gases is instructive.
In this chapter, we will study the general problem of a quantum-mechanical ideal
gas using the rules of quantum statistical mechanics developed in the previous chapter.
Following this, we will specialize our treatment for the fermionic and bosonic cases,
examine a number of important limits, and finally, derive the general concepts that
emerge from these limits.

11.2 General formulation of the quantum-mechanical ideal gas

The Hamiltonian operator for an ideal gas ofNidentical particles is

Hˆ=

∑N

i=1

pˆ^2 i

2 m

. (11.2.1)

In order to compute the partition function, we must solve for the eigenvalues of this
Hamiltonian. In so doing, we will also determine theN-particle eigenfunctions. The
eigenvalue problem for the Hamiltonian in the coordinate basis reads

−

̄h^2

2 m

∑N

i=1

∇^2 iΦ(x 1 ,...,xN) =EΦ(x 1 ,...,xN), (11.2.2)