The equipartition theorem:
a classical but sometimes useful result
Goal: understanding why in the classical limit the average energy associated with any
momentum or position variable appearing quadratically in one term of the
Hamiltonian* is ½ kBT
There is a more general form of the equipartition theorem which we don’t consider due to the limitations of
the classical approximation in first place
Example for usefulness:
cV-> 3R Dulong-Petit limit
Photons* and Planck’s black body radiation law
We start from a system with dynamics determined by Hamiltonian:
1
2
H  p1 , ..., p 3 N , q1 , ..., q 3 N   K u  H 
2
where u is one of the general coordinates q1, p1, · · · , q3N , p3N , and H’ and K
are independent of u
1
Next we show for :  ( u ) :
Ku
 (u ) 
2
1
2
2
k BT
The thermal average is evaluated according to:
 (u ) 

with
d
3N
d
qd
3N
p  (u ) e
3N
qd
3N
 du  ( u ) e
 du e
xu
pe
  H  p1 , ..., p 3 N , q1 , ..., q 3 N
  H  p1 , ..., p 3 N , q1 , ..., q 3 N
  (u )

  (u )


1 / 2
 dx e

1
2
Kx
2
d


ln  du e
dx  du

 
 (u )  
 ln 
 


  (u )
3N
d
qd
3N
3N
qd
3N
pe
  (u )
  (u )
e

1
2

1
2
k BT
 dx e

1
2
Kx
2
e
 H 
 H 


  1
 
  ln   ln
  2

 (u ) 
p  (u ) e



Let’s apply equipartition theorem to the Hamiltonian describing the lattice vibrations
of a solid in harmonic approximation
H 
1
2
3N

pj j qj
2
2
2
j 1
3N momentum variables appearing quadratically in H
3N position variables appearing
quadratically in H
6N degrees of freedom each contributing with
1
2
model independent classical limit
U  E  6N
1
2
k B T  3 N k B T  3 nN A k B T  3 nR T
 U 
CV  
  3 nR
  T V
Dulong-Petit limit
k BT
Equipartition theorem (classical approximation) is only valid when the typical
thermal energy kBT greatly exceeds the spacing between quantum energy levels:
Heat capacity of diatomic gases:
an example for application and limitation of the equipartition theorem
Let’s consider N non-interacting H2 molecules in harmonic approximation
N- molecule Hamiltonian H 
N
H
i
i 1
Single molecule Hamiltonian Hs reads
Hs 
p
2

1
2 m1
p
2
2
2m2
with m 1  m 2  m
  ( r 2  r1 )
for, e.g., H2 molecule
Here already we can see that in the harmonic approximation we will have 7
terms quadratically entering Hs
Hs 
1
2 m1

p1, x  p1, y  p1, z  
2
2
2
1
2m2
p
each molecule contributes with
2
2,x
7
2
 p 2 , y  p 2 , z   const r
k BT
2
2
2
For a closer look we separate out center of mass motion and introduce
spherical coordinates for the relative motion
Let’s start from the single molecule Lagrangian
Ls 
1
2
2
m1 r 1 
1
2
2
m2 r 2   ( r 2  r1 )
and introduce the center of mass
R 
m1 r 1  m 2 r 2
with M=m1+m2
M
and r  r 2  r 1
Ls 
1
m1 r 1  m 2 r 2
M
r  r 2  r1
2
M R 
2
R 
1
r  ( r )
2
2
where µ is the
reduced mass
r1  R 
r2  R 
 
m1 m 2
M
With L we derive the canonic conjugate momentum variables
 PX , PY , PZ 
 LS LS LS 

,
,

 X Y Z 
and
 LS LS LS 
p
,
p
,
p

 x y z  , , 
 x y z 
m2
M
m1
M
r
r
p   r
P  M R,
We express the Hamiltonian and Lagrangian in the new variables:
Hs 
P
2

2M
p
2
 ( r )
2
center of mass motion
With
x  r sin  co s 

2M
p
2
2
 ( r )
relative motion
spherical symmetric
Since    ( r  r )
express H rel 
Ls 
P
2
p
2
2
  ( r ) in spherical coordinates
x  r sin  cos    r cos  cos    r sin  sin 
y  r sin  sin 
y  r sin  sin    r cos  sin    r sin  cos 
z  r co s 
z  r cos    r sin 
Together with p r 
, p
 
L
2
r

, p

L
calculated from

1
1
2
2
2
2
2 2
2 2

  (r )    x  y  z    (r )    r   r   r    (r )
2
2
2
p
L rel
L
H rel 
1
2 r
1
2
p 
2
2
2  r sin 
2
2
p 
1
2
2
rotational contribution
pr   (r )
vibrational contribution
Total Hamiltonian:
2
H 
2
Px  Py  Pz
2M
3
2
k BT
2

1
2 r
1
2
2
p 
2
2  r sin 
2
k BT
2
p 
1
2
pr   (r )
2
 (r )  r
k BT
2
The activation of all degrees of freedom requires remarkable high temperatures
Large quantum effects whenever level spacing     k B T
not fulfilled
vibrations are
frozen out when
k BT  
