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222 7 QUANTUM THEORY
P7A.8 (a) As λ decreases, hc/λkT increases, and so ehc/λkT increases.�erefore, for
very short wavelengths, ehc/λkT is very large and 1 is negligible compared
to this. Hence,
lim
λ→0
ρ(λ, T) = 8πhc
λ5ehc/λkT = 8πhc
λ5
e−hc/λkT
(b) Comparisonwith the empirical expression gives the constants as a = 8πhc ,
and b = −hc/λ .
(c) �e total energy density at temperature T is given by [7A.7–240],
E(T) = ∫
∞
0
ρ(λ, T)dλ = ∫
∞
0
8πhc
λ5
e−hc/λkTdλ
Let x = hc/λkT , or λ = hc/xkT .�en, dλ = −hc/x2kT dx
E(T) = 8πhc∫
∞
0
1
hc/xkT
e−x dx
x2
= 8πhc ( kT
hc
)
4
∫
∞
0
x3e−x dx
= 8π(KT)
4
(hc)3
× 3! = 48πk
4
(hc)3
T4
�e integral is of the form of Integral E.3 with n = 3 and k = 1. �is is
consistent with the Stefan–Boltzmann law, as the energy density is pro-
portional to T4.
(d) �e energy spectral density is maximized at λ = λmax, where dρ/dλ = 0.
�is gives
dρ
dλ
= 8πhc d
dλ
(λ−5e−hc/λkT)
= 8πhc [dλ−5
dλ
e−hc/λkT + λ−5
de−hc/λkT
dλ
]
= 8πhc [−5λ−6ehc/λkT + λ−5 × hc
λ5kT
ehc/λkT]
= 8πhce
hc/λkT
λ7
[−5λ + hc
kT
]
�is expression equals 0 at λ = λmax, and is solved when
−5λmax +
hc
kT
= 0
λmaxT = hc
5k
�at is λmaxT is a constant, which is consistent with Wien’s law
P7A.10 �e Einstein temperature is given by θE = hνE/k; this has units of K (temper-
ature), as expected
(J s) × (s−1)
JK−1
= K