44 Gases: the density ofstates
the particleis confinedwithinanotherwise emptybox ofvolumeV?For conve-
nience,let us assume that the box is a cube of sidea(soV=a^3 ). Furthermore we
assume that ‘confined’ means thatit must satisfytheboundary conditionψ= 0 over
the surface ofthebox. Now we come to the ‘economy’referredto above: weknow
that the wavefunctionψis a simple sinusoidal oscillation inside the box. This simple
statementis a correct summary ofthesolution ofSchrödinger’s equationfor a particle
ofmassm,orofthe wave equationfor a soundor electromagnetic wave. Therefore
the only way to achieve a possible wavefunction is to ensure that a precisely integral
number ofhalf-wavesfitinto theboxinallthree principaldirections. Hence thetitleof
this section! The problem is the three-dimensional analogue of the one-dimensional
standing-wave normal modes of the vibrations of a string.
Tobe specificwechoose theoriginofco-ordinates tobe at one corner ofthebox.
The wavefunction is thengiven bya standingwave of the form
ψ∼sin(n 1 πx/a)·sin(n 2 πy/a)·sin(n 3 πz/a) (4.1)where the positiveintegersn 1 , 2 ,3(= 1 ,2,3, 4 ...)simply give the number ofhalf-
waves fittinginto the cubical box (sidea)in thex,yandzdirections respectively.
(The use of sine functions guarantees the vanishing ofψon the three faces of the
cube containingtheorigin; theintegralvalues ofthethreensensures the vanishing
ofψon the other three faces.) Incidentally, in terms of our earlier notation the state
specification by the three numbers(n 1 ,n 2 ,n 3 )is entirely equivalent to the previous
state label ‘‘j’.
It is useful to write (4.1) in terms of the components of a ‘wave vector’kas
ψ∼sin(kkxx)·sin(kkkyy)·sin(kkzz) (4.2)where
k=(kkx,kkky,kkz)=(π/a)(n 1 ,n 2 ,n 3 )The possible states for the wave are specified by giving the three integersn1,2,3,i.e. by
specifyinga particular pointkin‘k-space’. Thisisavaluablegeometricalidea. What
it means is that all thepossible ‘k-states’ can be represented (on a single picture) by
a cubic array of points in the positive octant ofk-space, the spacing being(π/a).
Nowbecause ofthe macroscopicsizeaofthebox, this spacingis verysmall inall
realistic cases. Indeed we shall find later that in most gases the states are so numerous
that ourNparticles are spreadover very many more thanNstates –infact most
occupation numbers are 0. Under these conditionsitisclear that we shallnot wishto
consider the states individually. That would be to specify the state of the assembly in
far too muchdetail. Ratherweshallchoose togroupthe states, andto specifyonly
the mean properties ofeachgroup. Andthe use ofk-spacegives animmediate way
of doing the grouping (or graining as it is often called).
Forinstancelet us suppose we wishto count the number ofstatesin a group
whoseklieswithx-componentbetweenkkxandkkx+δkkx,withy-componentbetween