Statistical Physics, Second Revised and Enlarged Edition

48 Gases: the density ofstates

4 .2.2 The dispersion relation

The second vital idea is that to make any progress with statistical physics one must

not onlybeable to count up the quantum states. One must alsoknow their energies,

εεjin earlier notation. Thepointissimplythat allequilibrium thermalproperties are

governed only by energy; the two constraints on an allowed distribution are those of

energy andnumber.

Inorder todetermine the energyεεj,there are two considerations. Thefirst ofthese

is the ‘plus’ of the previous section. If there are internal spatial degrees of freedom,

these willaffect the energy (see Chapter 7). In addition, sometimes the energyis

spin-dependent, for example in the case of an electron in a magnetic field, and then

this must be included and it will have important effects. But the second and more

generalconsiderationisthatin everycase the energy dependsonk.Weuseε(k),

or simplyε,in the rest of this chapter to represent the energycontribution from the

translational motion, i.e. from thek-state. Theε−krelation is often referred to as

thedispersion relation. It contains preciselythe same physicalcontent as theω−k

dispersion relation in wave theory(sinceε=ω)and theε−penergy-momentum

relation in particle mechanics (sincep=k).

Itis usuallya conveniencein statisticalphysics to combine thegeometryofk-

space (equation (4. 5 )) with theε−krelation togive a ‘densityof states in energy’,

often called simply the density of states. This quantity is defined so thatg(ε)δεis the

number ofstates withenergiesbetweenεandε+δε.It is derived from (4.5) byusing

the dispersion relation to transform both thek^2 factor and the rangeδk.Thispoint is

worthstressing. Itisalways worthwriting adensity ofstatesfunction withits range

explicitlyshown, as for example on both sides of (4.3), (4.4) and (4.5). Although

the same symbolgis usedforg(k)andforg(ε),these two functions have different

dimensions andunits. Itisthefunctions multipliedbytheir ranges,i.e.g(k)δkand

g(ε)δε,whichhave the samedimensions–theyare pure numbers. An exampleof

this transformation is given in the next section.

4.3 An example – heliumgas

Asaspecific example of the ideas of this chapter, let us consider helium (^4 He)gas
contained in a 10-cm cube at a temperature of 300 K.
The states aregivenbythesolution ofthetime-independent Schrödinger equation
for aparticle of massMin a field-free region

(−^2 / 2 M)∇^2 ψ=εψ (4.6)

withboundary conditions asdiscussedin section 4.1. Thesolutions are precisely

those of (4.1) and (4.2). Substitution of these solutions back into (4. 6 )gives for the

energies of the particle

ε=^2 k^2 / 2 M (4.7)