Statistical Physics, Second Revised and Enlarged Edition

Other informationfor statistical physics 47

The correspondingone-dimensionalresultfor aline (a ‘quantum wire’) oflength

Lis rather similar to (4.3) since no integration is needed

g(k)δk=L/( 2 π)δk

The reader might care to derive these results.

4.2 Other information for statistical physics

Inthe previous section wehavedescribedthegeometryoffittingwavesintoboxes,a
common feature for all types ofgaseous particle. However, before we know enough
about the states to proceed with statistical physics, there are two other ideas which
needtobestirredin. Althoughthefinalanswers will bedifferentfordifferentgases,
the same twoquestions need to be asked in eachparticular case.

4 .2.1 Quantum states arek-statesplus

The first question relates to what constitutes a quantum state for a particle. The general
answeristhat thefullstate specification (labelledsimply byjin earlier chapters) must
include a complete statement of what can inprinciple be specified about theparticle.
Thek-state of the particle (as given by the three numbersn1,2,3for our cubical box)
isafullspecification onlyofthe translationalmotion ofthe centre ofmass ofthe
particle, as characterized byψ(x,y,z).
But usuallythere are otheridentifiablethings one can say about the particle,besides
its centre ofmass motion. Firstly, one mayneedto considerinternalmotion ofthe
particle, arising from vibration or rotation of a molecular particle for example. We
shallreturn to this type of‘internaldegree offreedom’inChapter 7. Secondly, even
forasimple particle, one must specifythespinofthe particle, or thepolarization of
the corresponding wave, essentially the same idea. For example an electron witha
particular spatialwavefunction andits spin-upisinadifferentquantum statefrom
anelectron withthe same spatialwavefunctionbutits spin-down (hence theperiodic
table, for instance!). Similarly for an electro-magnetic wave one needs to know not
onlyits wave vectorkbut alsoits polarization (e.g.left- or right-handed).
Theidea ofspinmayeasily beincludedin (4.4). We redefineg(k)δkto be the
number of quantum states (and not merelyk-states) with magnitude ofkbetweenk
andk+δk.The equation thenbecomes

g(k)δk=V/( 2 π)^3 · 4 πk^2 δk·G (4.5)

where thenewfactorGisapolarization or spinfactor, usually1 or 2 or another small
integer, dependent on the type of substance under consideration. In future all density
ofstatesfunctionslikeg(k)willrefer to quantum states andwill includethisspin
factor.