1549380323-Statistical Mechanics Theory and Molecular Simulation

424 Quantum ideal gases

an electron from one of the occupied energy levels—is closely relatedto the Fermi
energy.

11.5.3 Zero-temperature thermodynamics

The fact that states of finite energy are occupied even at zero temperature in the
fermion gas means that the thermodynamic properties atT= 0 are nontrivial. Con-
sider, for example, the average particle number. In order to obtain an expression for
this quantity, recall that

〈N〉=

∑

m

∑

n

〈fnm〉=

∑

m

∑

n

θ(εF−εn) =g

∑

n

θ(εF−εn). (11.5.35)

In the thermodynamic limit, the sum may be replaced by an integrationin spherical
polar coordinates

〈N〉=g

∫

dnθ(εF−εn)

= 4πg

∫∞

0

dn n^2 θ(εF−εn). (11.5.36)

However, since the energy eigenvalues are given by

εn=

2 π^2 ̄h^2

mL^2

n^2 , (11.5.37)

it proves useful to change variables of integration fromntoεnusing eqn. (11.5.37):

n=

(

mL^2

2 π^2 ̄h^2

) 1 / 2

ε^1 n/^2

dn=

1

2

(

mL^2

2 π^2 ̄h^2

) 1 / 2

ε−n^1 /^2 dεn. (11.5.38)

Inserting eqn. (11.5.38) into eqn. (11.5.36), we obtain

〈N〉= 4πg

∫∞

0

dn n^2 θ(εF−εn)

= 2πg

(

mL^2

2 π^2 ̄h^2

) 3 / 2 ∫∞

0

dεnε^1 n/^2 θ(εF−εn)

= 2πg

(

mL^2

2 π^2 ̄h^2

) 3 / 2 ∫εF

0

dε ε^1 /^2