1000 Solved Problems in Modern Physics

280 4 Thermodynamics and Statistical Physics

4.46 For stationary waves, in thex-direction
kxa=nxπ
ornx=kxa/π
dnx=(a/π)dkx
Similar expressions are obtained foryandzdirections.
dn=dnxdnydnz
=(a/π)^3 d^3 k
However only the first octant of number space is physically meaningful.
Therefore
dn=(1/8)(a/π)^3 d^3 k
Taking into account the two possible polarizations

dn=

2 V

(2π)^3

d^3 k=

2 V

8 π^3

. 4 πk^2 dk

Butk=

ω

c

;dk=dω/c

∴dn=

Vω^2 dω

π^2 c^3

4.47n!=n(n−1)(n−2)...(4)(3)(2)

Take the natural logarithm of n!

lnn!=ln 2+ln 3+ln 4+···+ln(n−2)+ln(n−1)+lnn

=Σnn= 1 lnn

=

∫n

1

lnndn

=nlnn−n+ 1

≈nlnn−n

where we have neglected 1 forn 1

4.48 p(E)=(2J+1)e−J(J+1)

(^2) / 2 ikT
The maximum value ofp(E) is found by setting dp(E)/dJ= 0
[
2 −

(2J+1)^2 ^2

2 I 0 kT

]

e−J(J+1)

(^2) / 2 I 0 kT
= 0
Since the exponential factor will be zero only forJ=∞,
[
2 −

(2J+1)^2 ^2

2 I 0 kT

]

= 0

Solving forJ, we get

Jmax=

√

I 0 kT



−

1

2