Bose systems: photons, phonons & magnons
Photons* and Planck’s black body radiation law
*For a formal derivation of photons via quantization of the EM field see, e.g., http://en.citizendium.org/wiki/Quantization_of_the_electromagnetic_field
Ekin
max
  
Einstein Nobel prize 1921
Existence of photons
E 
Plot from
Our goal: Using photon interpretation to
understand
u( )d
2 with
=frequency
Planck’s const./2
Energy of the quantum
Energy per unit volume
of the radiation emitted in the frequency range [, +d]
Standing EM waves = modes characterized by wave vector k
of plane wave solutions  eik r
Each mode can be occupied with a certain # nk,s of photons
nk,s of photons  classical amplitude of the standing wave
s=1,2 takes into account the two linear independent polarizations
With this and the individual photon energies k , s
1

Total energy in the cavity E   k ,s  nk ,s     k ,s nk ,s  E0
2  k ,s

k ,s

Index of a particular microstate characterized by the occupation #s ..., nk , s ,...
E    j n j  E0
with j   k , s 
j
Let’s first take advantage of the fact that we know the BoseEinstein distribution function already
U  E    j n j  E0  
j
j
j
e
 j
1
 E0

We derive the same result with
the partition
function of the canonical ensemble
Energy
fluctuations
Z  e
  E

 e  E0  e


jnj
 e   E0
j


e
   1n1  2 n2 ...
n1 , n2 ,..
E    j n j  E0
j
1

 e  E0 

 1 e
1
1



 1  e
2
1



 1  e
3
With


Z


1


  E0
...

e





j  1 e
  E
E
e
 
j



1 Z
 ln Z


  E


e
Z 



 F 
S


Of course consistent with U=F+TS when using F=-kBT lnZ and


T V

 ln Z
With U  
  E
  E
e


E
e

 
U



 

U

E

ln
1

e
 0 
 
j

j




 E0  
j
 je
1 e
  j
  j
 E0  
j
j
e
 j
1
The dispersion relation for light propagating in vacuum reads:
k ,s  c k
speed of light
Reveals: light frequency is independent of the polarization state s and the
direction of propagation
U  E0  
j
j
e
 j
1
 E0  
k ,s
k ,s
e
 k ,s
1
 E0  2
k
k
e
 k
1
Let’s evaluate the k-sum
If we impose periodic boundary conditions for the electric field wave
E  E0 e 
i k r t 
such that
   Lx   
   
i  k  r   Ly   
   L 
   z 
kx Lx  2 nx
2


i k r 

k

,

k

,

k

k y Ly  2 ny
x
y
z
e
e
L
L
Lz
x
y
kz Lz  2 nz
Lx Ly Lz
1
V
2
...

dk
dk
dk
...

dk
dk
dk
...

4

k
dk ...
k
x
y
z
x
y
z
3 
3 

k x k y k z
 2 
 2 
V 4 k 2 dk
 2 
# of modes between k and k+dk
Density of states in k-space
 D (k )dk
3
We will learn other techniques to calculate the density of states for
more complicated dispersion relations.
U  E0  2
k
 E0  2
k
e
 k
1
 E0  2 D(k )
0
4 k

 2 
2
3
e
0
With:
 (k )  c k
d
dk 
k

V
U  E0 
 k
1

V

u
c
3
2
3
 e   1
d
3
2

e
 k

2
 
1
with x   
0


x3
x3
4
u  3 3 4 2  x dx with  x dx 
c   0 e 1
e 1
15
0
1
1
dk
dk
c 0 e
U  E0
Defining the energy density u 
V
c
k

d
Internal energy of the box at T, relative to the
vacuum energy (e.g., responsible for the Casimir effect)
d 
dx


u   u ( )d 
0
 2 kB4
15c3
3
T4
If we asked for the spectrally resolved u( )d
we obtain from inspection of
u
 2 
c3 2

u
3
 e    1 d
c 0
3
8h
3
0 e h  1 d  c3 0 e h  1 d
4 

3
2
8h  3
u ( )d  3  
d Planck’s spectral
energy density
c e 1
Image from: http://en.wikipedia.org/wiki/File:Max_Planck.png
It is useful to write it down with the differential to remember how to do the correct
transformation to u ( )d 
8h 
u ( )d  3  
d
c e 1
3
u ( ) d  
8h c

5
1
e
 hc / 
1
d
For experimental comparison we need to know not what is the spectral energy
density insight the box, but what radiates out into one hemisphere:
z
Photons leaving the hole in direction 
in the time t come out of max depth of

c cos  t
A
c cos 

c cos  t
c
#Photons leaving the hole in direction  in time t  A c cos  t
Fraction of photons in solid angle d=sindd is
d  / 4
Energy emitted into hemisphere in time t
Act
U
V
 /2
With sin=x

0
2
1
Act
d  d
cos sin   U
4
2V
0
 /2
 d cos sin 
0
Act
Act
Act
U
x dx  U
u

2V 0
4V
4
1
Lambert’s
cosine law*
The emitted intensity is defined emitted energy per area A and time t
I u
With
u
Act
c
u
4 At
4
 k
2
15c3
4
B
3
I
T4
 2 k B4
60c 2
3
T 4  T 4
Celebrated Stefan-Boltzmann law
A second look at the derivation of Planck’s law
k

U  E0  2 D(k )
0
e
 k
1

dk  2 d D()
0
# of modes with frequency [,+d ]
1
e
 
1

Photon energy
# of photons excited in each mode
The concept of density of states
Continuous volume element in isotropic k-space
D(k )dk 
V
 2 
3
4 k 2dk
Region in k-space occupied by a quantized state:
Lx Ly Lz
k x k y k z 
2 2 2
k
y
2
L
k
x
From density of states in k-space to density of states in energy or frequency space
D(k )dk 
V
 2 
 (k )  c k
3
4 k 2dk
k

With dispersion relation, here for photons
and dk 
c
Simple substitution yields
d
c
D( )d 
   d
2
V
 2 
However, k-space not always isotropic
4  
c
3
c
more general approach
D( )   (   (k )) Property: when integrating over [, +d ] we obtain the
k
number of states in this interval
 (k )  c k
4V
V
2
2


 (  (k ))dk
4

k

(



(
k
))
dk
D( )   (   (k )) 
k
3 
3 
 2 c 
 2 
k
Here
D( ) 
4 V  2
 2 c 
3