Statistical Physics, Second Revised and Enlarged Edition

46 Gases: the density ofstates

Inthis method,theperiodic conditionisthatbothψanditsgradient should

match up on opposite faces of the cube. (One way of picturing this is that if the

wholeofspace werefilledwithidenticalboxes, the wavefunction would be smooth

andwould be exactlyreplicatedin eachbox.) The normalmodes are then travelling

waves of the form

ψ∼exp(ikkxx+ikkkyy+ikkzz)

or

ψ∼exp(ik·r)

The values ofkdiffer from the earlier ones in two respects, and indeed the
microscopicpictureisquitedifferentindetail.Thefirstdifferenceisthat the
spacingofthek-valuesisdoubled,since now theboundarycondition requires
fitting an integral number of full (and not half) waves into the box. Hence the
spacingbetween allowedk-valuesbecomes ( 2 π/a), and(4.3)isnolonger valid–
another factor of(^18 ) is needed on the right-hand side. The second difference is
that the restriction onkto the positive octant is lifted. A negativek-value gives
a different travellingwave statefrom positivek;theyrepresent states withthe
same wavelength but travellingin opposite directions. This difference means that
the integration between (4.3) and (4.4) should cover all eight octants ofk-space.
Therefore the twodifferences compensatefor eachotherin workingoutg(k)δk
and the result (4.4) survives unchanged.
3. The next comment concerns graining. The states are not evenly spread ink-space
onthefinestlevel–theyarediscrete quantum states. Therefore we cannotinprin-
ciple let the rangeδkin (4.4) tend to zero. The group of states we are considering
must alwaysbealarge onefor the concept ofthe (average)density ofstates to make
sense. Nevertheless,in everypracticalcase except one (the Bose–Einstein con-
densation to be discussed in Chapter 9) we shall find that the differences between
adjacent groups can still besominutely small,that calculus canbe used,i.e. we
canfor computation replace thefinite rangeδkbytheinfinitesimaldk.

4. The final comment is about dimensionality. Equation (4.4) is a three-dimensional
   result,basedon thevolume ofasphericalshellofradiusk. Entirely analogous
   results canbederivedforparticles constrainedwithin one- or two-dimensional
   ‘boxes’, a topic of much importance to modern nanoscience. For example consider
   the states ofa particle constrainedin a two-dimensionalsheet. The confinement
   happensbecause thewidthofthebox normalto thesheetismadesosmallthat
   the wavefunction in that direction is fixed to be lowest standing wave state (one
   halfwave across thesheet), withthe next state (twohalfwaves) out ofthermal
   energyrange. The wavefunctions within the sheet can again be treated as extended
   travelling waves. The particles in a sheet of areaAhave a density of states inkof

g(k)δk=A/( 2 π)^2 · 2 πkδk