fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
(e) Planck’s Theory of Radiation (Quantum Mechanical Theory)
Basic Assumption and Methodology
(1) Max Planck used Boltzmann distribution to determine the average energy of black
body radiation. The probability to find energy between
E
to
E  dE
is

 E
1
P  E  dE 
exp  
k BT
 k BT
 E P  E dE

0
.
 dE . The average energy is given by E  

 P  E dE
0
(2) Planck assumed that energy E is not continuous rather discrete values. According to
Planck, the allowed energy for black body radiation for frequency  is E  nh where
n  0,1, 2,3...
(3) The average energy is E 
 EP  E 
n 0
 PE
n 0
 nh
nh
exp  
n  0 k BT
 k BT

 nh
1
exp  

n  0 k BT
 k BT



h


 h
 exp 

 k BT

 1

Derivation of Average Energy

Put the value,  
h
,
kT
E  k BT
 n exp  n 
0

 exp   n 
0


d
ln  exp   n  
d n  0



d 
d
exp   n   n exp  n 
 exp   n    
d n 0
 n 0 d
 n 0


 exp   n 
 exp   n 
n 0
n 0

 exp  n 
n 0


d
d


E  k BT  
ln  exp   n     h
ln  exp  n 
d n 0
d 0



2
3
 exp   n   1  exp     exp  2   exp  3  ...  1  X  X  X  ...
n 0
where X  exp   
1  X  X 2  X 3  ...  1  X 
1
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Revised Edition 2019
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fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
E  h
1
d
ln 1  exp    
d
 h
1  exp    
1
 1 1  exp    
2
exp    
h
 h
exp 
 k BT

 1

(3) Planck also used the same analogy of Rayleigh and Jeans theory to count the number
8 Va 3 2
of standing waves dN   in the frequency interval  to   d dNd 
d
c3
(4) Energy density    d between frequency  to   d is    d  N   d E
   d 
8 2
.
c3
Rayleigh - Jeans
h
 h 
exp 
 1
 k BT 
Wein's Distribution
ρ(v)dv
Planck's Distribution
Observed

Figure 1: Energy density versus frequency according to Planck
The equation does accurately describe the low frequency (high wavelength) spectrum of
Frequency,vHz
thermal emission from objects, as well as accurately fit the experimental data for high
frequency (short wavelengths) emission.
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
Revised Edition 2019
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