Use the gamma function, explicitly indicating that E) = EN2
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Transcribed Image Text:Consider the system in the limit in which the energy E of the gas is much greater
than the ground-state energy, so that all of the quantum numbers are large. The
classical version of statistical mechanies, in which we divide up phase-space into
cells of equal volume, is valid in this limit. The mumber of states (E,V) lying
For an ideal gas, the total energy E does not depend on the positions of the
particles (see Eq. (104). This means that the integration over the position vectors
T, can be performed immediately. Since each integral over ri extends over the
volume of the container (the particles are, of course, not allowed to stray outside the
between the energjes E and E + SE is simply equal to the number of cells in
container). f d'r, = V. There are N sosh integrals, so Eq. (105) reduces to
phase-space contained between these energies. In other words, 2(E,V) is
ALE,V) « V*x(E).
(108)
proportional to the volume of phase-space between these two energies:
(105)
where
X(E) x
(109)
Here, the integrand is the element of volume of phase-space, with
dr, dy, dz,
(106)
is a momentum space integrai which is independent of the volume.
d'p - dp.. dp., dp.
The energy of the system can be written
(107)
E =
(110)
where (E, y. z) and (pa- Piy. Pia) are the Cartesian coordinates and
2m
lanl
momentum components of the ith particle, respectively. Tbe integration is over all
coordinates and momenta such that the total energy of the system lies between E
and E + 6E.
Proot
ME) = EN2
Use the gamma function, explicitly indicating that (E) = EN2
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