11
Applications III
Quantum Domain
Quantum behavior dominates at the atomic and molecular levels although classical
models are usually sufficiently accurate at moderate to high temperatures. Conversely,
quantum effects become more noticeable at low temperatures. Here we apply techniques
introduced in the last chapter to quantum models of solids and radiation. These
applications are singularly important.
Application: Simple Models of Solids
A solid material was profitably viewed as an assembly of classical harmonic oscillators
with a common frequency . Applying the equipartition theorem, we find a heat capacity of 3Nk
for solids. This is fairly accurate for most solids at room temperature and is known as the law of
Dulong and Petit. However, at low temperatures the heat capacity diminishes and a better model
is needed.
Einstein proposed a quantum mechanical version of the oscillator model and this gave a
better description of solids at low temperatures. A one-dimensional quantum mechanical
oscillator with a frequency  has energy levels En given by
(1)
En  n   21 
where n is an integer. We can ignore the ground state energy (  2 ) and construct the partition
function Z1,x.,

Z1, x   exp n   .
(2)
n0
This series can be written in closed form using the following:
1
(3)
1  r  r 2  r 3  
1 r
(See a derivation at the end of this chapter.) Once the partition function is constructed, energy
and heat capacity are easily found as prescribed in the following problem.
7.
(a) Develop a one-particle, one-dimensional partition function for a quantum mechanical
1
harmonic oscillator in closed form. [ans. 1  exp   ]
(b) Use the result of (a) to construct the partition function for N oscillators in three
dimensions. (Einstein model of a solid.)
(c) Show the average energy for this solid is E 
3 N
exp kT   1
(d) Find the high and low temperature limits of heat capacity. Notice that this agrees
with a classical oscillator model at high temperature, but deviates at low temperature.
Debye introduced a model that does not assume the oscillators are independent and is
2
more accurate at low temperatures. [ans. 3Nk , 3Nk  exp(  / kT ) /( kT )2 ]
Applications III
2
Occupation Numbers: Two Kinds of Identical Particles
Quantum physics reveals that there are two kinds of identical particles; a specific
quantum state can be occupied by only one Fermi particle but by any number of Bose-Einstein
particles. For example, electrons are fermions and no two can occupy the same spin and angular
momenta in a hydrogen atom. Photons are bosons and a limitless number can occupy the same
space. We will present simple computations to produce the following results for the average
number nj that occupy a particle energy state Ej:
nj 
1
 Ej 
exp   1
 kT 
nj 
1
 Ej 
exp   1
 kT 
Fermi Particles
(4)
Bose Particles
(5)
 
Writing w nE j to represent the canonical probability for n particles each with energy Ej,
the above expressions are easily computed by averaging the numbers 0 and 1 for fermions and
numbers 0, 1, 2,, for bosons.
n j  0  w0  1 w 1E j Fermi particles
nj
 
 0  w0  1 w1E   2  w2 E 
j
j
Bose particles
1.
Derive the average occupation number for a Fermi particle in energy level Ej: What are
the high and low temperature limits for this number?
[ans. ½ for high temperature and exp( E j ) for low temperature]
2.
Derive the average occupation number for a Bose particle in energy level Ej. (The series

1
of Eq.(16) can be used here. Then show that n j   ln
). What are the
 1  exp( E j )
high and low temperature limits for this number?
[ans. kT / E j and exp( E j ) ]
Application: Blackbody Radiation
Blackbody radiation is an extremely important application of statistical mechanics. A
heated object radiates energy in the form of photons whose frequency distribution depends on the
temperature of the object. Photons are Bose particles and therefore obey the occupation number
expression (5). Each photon contributes energy of hf or  to the radiation and this is related to
the momentum magnitude p by

(6)
p
c
where c is the speed of light.
Applications III
3
The following problem highlights the primary results for
radiation. Two technical details should be mentioned. First, the
integration over frequency (or over number) must be restricted to
positive values. This means that instead of integrating over an
entire sphere of radius |p| (or |n|), we are limited to one octant; that
is, an area of 81 4p 2 as shown in the diagram (where x,y,z
represents either p x , p y , p z .or n x , n y , n z ). A second particular is
that photons can have two polarizations, so the multiplicity of
states is two.
3.
(a) Show that a differential expression for the number of photons in an angular frequency
V
2d
range d is given by dN   2 3
. (A factor of 2 is included because
 c exp  / kT   1
there are two independent directions of polarization and a factor of 1/8 is included to
exclude negative frequencies.)
(b) Calculate the energy of the blackbody using
 dN

and show it is proportional to

4
VT . Use the integral
x 3dx  4
 e x  1  15 .
0
[ans. U  aVT 4 where a 
k 4 2
]
15(c) 3
(c) Find the heat capacity at constant volume CV from the result of part (b). Use this to
C dT
4
calculate entropy from the thermodynamic form dS  V
. [ans. S  aVT 3 ]
T
3
P
 S 
(d) Show that 
  and derive the state equation for radiation PV  13 U .
 V U T
(e) Use the result of part (c) to show the adiabatic condition for radiation is
V T 3  constant.
(f) The early universe was much smaller and hotter than the present 2.7K. when the
temperature fell to about 3000K, matter and the cosmic radiation became decoupled.
Assuming adiabatic expansion, by what fraction has the universe increased in volume
since the decoupling of cosmic radiation and matter?
[ans. ~1.37109 ]
hf
hc
x3

where x 
. Find
x
kT kT
e 1
the peak contribution to the radiation integrand as follows: Differentiate u with respect to
d
dF ( x ) dx
F ( x) 
 0 . Show this is satisfied by
 and set the result equal to zero. Use
d
dx d
the numerical solution x = 2.814 and derive the Wein displacement law
hc
2.898  103 m
 peak 

.
2.814kT
T
4 Part (b) of the last problem included an integrand u 
Applications III
4
5 Our sun has a maximum emission at a wavelength of approximately 5 x 10-7 m (500 nm).
Find the surface temperature of the sun. [ans. 5800 K]
x3
ex  1
Summary Remark
The notable quantum features of this chapter were (i) energy levels for the oscillator are
discrete (as they are for other stationary states) and (ii) particle indistinguishably is a quantum
phenomenon that is best treated by introducing occupation numbers.
Appendix to Chapter 11
Here is a derivation of the series of Eq.(3):
Let Z  1  r  r 2   and note that rZ  Z  1 . Solve for Z to find
1
Z
.
1 r