Quantum Physics, Engineering, Semester, by Prof. JK Goswamy Notes - Computer Science Engineering (CSE) PDF Download

 Page 1


 
 
 
 
QUANTUM PHYSICS 
 
 
1. Origin of Quantum Physics. 
2. Wave Particle Duality. 
3. Schrodinger’s Formulation. 
4. Applications of Schrodinger’s Equation. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 64 
 
Page 2


 
 
 
 
QUANTUM PHYSICS 
 
 
1. Origin of Quantum Physics. 
2. Wave Particle Duality. 
3. Schrodinger’s Formulation. 
4. Applications of Schrodinger’s Equation. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 64 
 
ORIGIN OF QUANTUM PHYSICS 
 
6.1 Black Body Radiation Distribution 
6.1.1 Features of Black Body Spectrum 
Every substance emits electromagnetic radiations, the 
character of which depends upon the nature and 
temperature of the substance. Some of the important 
features common to all bodies are: (i) When a body is 
placed in surrounding medium which is at a different 
temperature than that of body, then heat exchange 
takes place between the body and medium bilaterally. 
In this process, hot body (or medium) gets cooled 
while other gets hotter. This process continues till 
thermal equilibrium is reached. (ii) The bodies which 
are good emitter in their hot state are good absorbers 
in their cold state. (iii)  A perfect black body is one, 
which absorbs all the incident radiation at low 
temperatures and emits radiation of all the possible 
wavelengths, when heated to a high temperature. 
 
6.1.2 Ferry’s Black Body 
The common example for a practical demonstration of 
black body is the one devised by Ferry (see figure 6.1).  
 
Figure 6.1: The schematic view of Ferry’s Black Body. 
 
It consists of a spherical hollow metallic enclosure 
whose inner surface is coated with lamp black. This 
enclosure has a fine orifice on its surface and a conical 
projection on inner surface diametrically opposite to 
the orifice. The EM radiations entering the orifice, 
strike the projection and further get absorbed due 
multiple reflections through inner surface. In its hot 
state, radiations of all possible wavelengths are 
emitted through the fine orifice. 
 
6.1.3 Black Body Spectrum 
The distribution of energy among various wavelengths 
emitted is a function of temperature only irrespective 
of shape and size of the black body. The spectral 
energy density u(?)d? (the rate of energy density of 
radiation having frequency between ? and ?+d?) are 
plotted as a function of frequency of emitted 
radiation, then following graphical behavior (see 
figure 6.2), called black body radiation spectrum, is 
observed. The following are interesting features 
observed in this radiation energy distribution 
spectrum: 
6.1.3.1: Continuous Energy Distribution 
At each temperature, the black body emits radiations 
of all possible frequencies between minimum and 
maximum limits. This spectral energy distribution is 
continuous and non-uniform. The energy associated 
with a particular frequency increases with rise in 
temperature of the black body. The maximum 
possible frequency emitted by the black body 
increases with temperature of the black body. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 65 
 
Page 3


 
 
 
 
QUANTUM PHYSICS 
 
 
1. Origin of Quantum Physics. 
2. Wave Particle Duality. 
3. Schrodinger’s Formulation. 
4. Applications of Schrodinger’s Equation. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 64 
 
ORIGIN OF QUANTUM PHYSICS 
 
6.1 Black Body Radiation Distribution 
6.1.1 Features of Black Body Spectrum 
Every substance emits electromagnetic radiations, the 
character of which depends upon the nature and 
temperature of the substance. Some of the important 
features common to all bodies are: (i) When a body is 
placed in surrounding medium which is at a different 
temperature than that of body, then heat exchange 
takes place between the body and medium bilaterally. 
In this process, hot body (or medium) gets cooled 
while other gets hotter. This process continues till 
thermal equilibrium is reached. (ii) The bodies which 
are good emitter in their hot state are good absorbers 
in their cold state. (iii)  A perfect black body is one, 
which absorbs all the incident radiation at low 
temperatures and emits radiation of all the possible 
wavelengths, when heated to a high temperature. 
 
6.1.2 Ferry’s Black Body 
The common example for a practical demonstration of 
black body is the one devised by Ferry (see figure 6.1).  
 
Figure 6.1: The schematic view of Ferry’s Black Body. 
 
It consists of a spherical hollow metallic enclosure 
whose inner surface is coated with lamp black. This 
enclosure has a fine orifice on its surface and a conical 
projection on inner surface diametrically opposite to 
the orifice. The EM radiations entering the orifice, 
strike the projection and further get absorbed due 
multiple reflections through inner surface. In its hot 
state, radiations of all possible wavelengths are 
emitted through the fine orifice. 
 
6.1.3 Black Body Spectrum 
The distribution of energy among various wavelengths 
emitted is a function of temperature only irrespective 
of shape and size of the black body. The spectral 
energy density u(?)d? (the rate of energy density of 
radiation having frequency between ? and ?+d?) are 
plotted as a function of frequency of emitted 
radiation, then following graphical behavior (see 
figure 6.2), called black body radiation spectrum, is 
observed. The following are interesting features 
observed in this radiation energy distribution 
spectrum: 
6.1.3.1: Continuous Energy Distribution 
At each temperature, the black body emits radiations 
of all possible frequencies between minimum and 
maximum limits. This spectral energy distribution is 
continuous and non-uniform. The energy associated 
with a particular frequency increases with rise in 
temperature of the black body. The maximum 
possible frequency emitted by the black body 
increases with temperature of the black body. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 65 
 
 
Figure 6.1: The black body radiation distribution is 
shown as function of emitted wavelength and also as 
a function of temperature. 
 
6.1.3.2: Wien’s Displacement Law 
The frequency corresponding to which the spectral 
energy density (emitted by the black body) is 
maximum is called frequency of maximum emission. 
The corresponding wavelength, called the wavelength 
of maximum emission, shifts towards smaller values 
when the temperature of the black body is raised. This 
is called Wien’s displacement law. It is mathematically 
expressed as  
 ) 1 . 6 ( b T
m
= ? 
Where b = 2.898x10
-3
mK is called Wien’s constant. 
 
6.1.3.3: Fifth Power Law 
The energy emitted by the black body, corresponding 
to the wavelength of maximum emission, varies as 
fifth power of the absolute temperature of the black 
body. This is called Fifth Power Law. 
 
 
6.1.3.4: Stefan-Boltzmann Law 
The area under the curve represents the energy 
emitted per unit area per second by the black body 
corresponding to all the frequencies emitted by it. 
This area is found to be proportional to the fourth 
power of the absolute temperature of the black body. 
The Stefan’s law states that the energy emitted per 
unit area per second corresponding to all the 
wavelengths emitted by the black body is directly 
proportional to the fourth power of the absolute 
temperature. Mathematically  
 ) 2 . 6 (
4
T E s = 
The above formula gives the energy emitted by the 
black body per unit area per second when it is 
surrounded by black bodies at 0
o
K. Here s=5.67 x 10
-8
 
Wm
-2
K
-4
 is called Stefan’s constant. If the black body is 
surrounded by black bodies at temperature T
o
, then 
this law takes the form given as: 
) 3 . 6 ( ) (
4
0
4
T T E - = s 
If the body is not perfectly black and has emissivity e, 
then  
 ) 4 . 6 ( ) (
4
0
4
T T E - = es 
 
6.1.4 Rayleigh Jeans Treatment 
The shape of the black body radiation distribution 
curve is shown in the figure 6.2. This spectral energy 
distribution is dependent solely on the temperature of 
the black body and does not depend upon shape or 
dimensions of the body. Rayleigh and Jean made a 
theoretical attempt to explain the shape of the curve 
purely on the considerations of classical physics.  
The black body can be simulated as a metallic cavity 
having a fine hole in it and whose inner walls are 
perfect reflectors. The incident radiations, falling on 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 65 
 
Page 4


 
 
 
 
QUANTUM PHYSICS 
 
 
1. Origin of Quantum Physics. 
2. Wave Particle Duality. 
3. Schrodinger’s Formulation. 
4. Applications of Schrodinger’s Equation. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 64 
 
ORIGIN OF QUANTUM PHYSICS 
 
6.1 Black Body Radiation Distribution 
6.1.1 Features of Black Body Spectrum 
Every substance emits electromagnetic radiations, the 
character of which depends upon the nature and 
temperature of the substance. Some of the important 
features common to all bodies are: (i) When a body is 
placed in surrounding medium which is at a different 
temperature than that of body, then heat exchange 
takes place between the body and medium bilaterally. 
In this process, hot body (or medium) gets cooled 
while other gets hotter. This process continues till 
thermal equilibrium is reached. (ii) The bodies which 
are good emitter in their hot state are good absorbers 
in their cold state. (iii)  A perfect black body is one, 
which absorbs all the incident radiation at low 
temperatures and emits radiation of all the possible 
wavelengths, when heated to a high temperature. 
 
6.1.2 Ferry’s Black Body 
The common example for a practical demonstration of 
black body is the one devised by Ferry (see figure 6.1).  
 
Figure 6.1: The schematic view of Ferry’s Black Body. 
 
It consists of a spherical hollow metallic enclosure 
whose inner surface is coated with lamp black. This 
enclosure has a fine orifice on its surface and a conical 
projection on inner surface diametrically opposite to 
the orifice. The EM radiations entering the orifice, 
strike the projection and further get absorbed due 
multiple reflections through inner surface. In its hot 
state, radiations of all possible wavelengths are 
emitted through the fine orifice. 
 
6.1.3 Black Body Spectrum 
The distribution of energy among various wavelengths 
emitted is a function of temperature only irrespective 
of shape and size of the black body. The spectral 
energy density u(?)d? (the rate of energy density of 
radiation having frequency between ? and ?+d?) are 
plotted as a function of frequency of emitted 
radiation, then following graphical behavior (see 
figure 6.2), called black body radiation spectrum, is 
observed. The following are interesting features 
observed in this radiation energy distribution 
spectrum: 
6.1.3.1: Continuous Energy Distribution 
At each temperature, the black body emits radiations 
of all possible frequencies between minimum and 
maximum limits. This spectral energy distribution is 
continuous and non-uniform. The energy associated 
with a particular frequency increases with rise in 
temperature of the black body. The maximum 
possible frequency emitted by the black body 
increases with temperature of the black body. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 65 
 
 
Figure 6.1: The black body radiation distribution is 
shown as function of emitted wavelength and also as 
a function of temperature. 
 
6.1.3.2: Wien’s Displacement Law 
The frequency corresponding to which the spectral 
energy density (emitted by the black body) is 
maximum is called frequency of maximum emission. 
The corresponding wavelength, called the wavelength 
of maximum emission, shifts towards smaller values 
when the temperature of the black body is raised. This 
is called Wien’s displacement law. It is mathematically 
expressed as  
 ) 1 . 6 ( b T
m
= ? 
Where b = 2.898x10
-3
mK is called Wien’s constant. 
 
6.1.3.3: Fifth Power Law 
The energy emitted by the black body, corresponding 
to the wavelength of maximum emission, varies as 
fifth power of the absolute temperature of the black 
body. This is called Fifth Power Law. 
 
 
6.1.3.4: Stefan-Boltzmann Law 
The area under the curve represents the energy 
emitted per unit area per second by the black body 
corresponding to all the frequencies emitted by it. 
This area is found to be proportional to the fourth 
power of the absolute temperature of the black body. 
The Stefan’s law states that the energy emitted per 
unit area per second corresponding to all the 
wavelengths emitted by the black body is directly 
proportional to the fourth power of the absolute 
temperature. Mathematically  
 ) 2 . 6 (
4
T E s = 
The above formula gives the energy emitted by the 
black body per unit area per second when it is 
surrounded by black bodies at 0
o
K. Here s=5.67 x 10
-8
 
Wm
-2
K
-4
 is called Stefan’s constant. If the black body is 
surrounded by black bodies at temperature T
o
, then 
this law takes the form given as: 
) 3 . 6 ( ) (
4
0
4
T T E - = s 
If the body is not perfectly black and has emissivity e, 
then  
 ) 4 . 6 ( ) (
4
0
4
T T E - = es 
 
6.1.4 Rayleigh Jeans Treatment 
The shape of the black body radiation distribution 
curve is shown in the figure 6.2. This spectral energy 
distribution is dependent solely on the temperature of 
the black body and does not depend upon shape or 
dimensions of the body. Rayleigh and Jean made a 
theoretical attempt to explain the shape of the curve 
purely on the considerations of classical physics.  
The black body can be simulated as a metallic cavity 
having a fine hole in it and whose inner walls are 
perfect reflectors. The incident radiations, falling on 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 65 
 
this hole, get absorbed following multiple reflections 
inside the cavity. This cavity, when heated, results in 
excitation or even ionization of metal atoms which 
subsequently result in emission of radiation. The 
spectral distribution of radiation, emitted by such a 
body in its hot state, has similar features as that of a 
black body. The emitted radiation suffer multiple 
reflections through walls of the cavity and superpose 
to form standing waves. These standing waves, having 
nodes at inner surface of cavity, oscillate in different 
modes thereby resulting in radiations of all possible 
wavelengths or frequencies. These form so called the 
cavity radiations.  
The electromagnetic radiations inside such a cavity, at 
temperature T, were considered to be three-
dimensional standing waves. The condition for 
standing waves in such a cavity is that the path length 
from wall to wall, in whatever direction, must be 
integral multiple of half wavelengths and hence a 
node occurs at the reflecting surface. If the cavity is 
considered to be a cube of edge L, then standing 
waves must satisfy the following conditions in x, y and 
z directions. 
) 5 . 6 (
,... 3 , 2 , 1
2
,... 3 , 2 , 1
2
,... 3 , 2 , 1
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= =
= =
= =
z z
y y
x x
n
L
n
n
L
n
n
L
n
?
?
?
 
For standing waves in arbitrary direction, we must 
have an independent condition that: 
) 6 . 6 (
2
2
2 2 2
?
?
?
?
?
?
= + +
?
L
n n n
z y x
 
In expression (6.8), the quantity L denotes the path 
length in the arbitrary direction and not the edge 
length of cubical enclosure.  
Let’s calculate the number of standing waves n( ?)d ? 
per unit volume having wavelength between ? and 
?+d ?. This problem is same as determining the 
number of triplet sets (n
x
, n
y
, n
z
) that yield 
wavelengths in the interval ? and ?+d ? and then 
divide this by volume L
3
 of the cavity. 
For this purpose we can imagine an integer space (see 
figure 6.3) whose coordinate axes are n
x
, n
y
 and n
z
 
respectively and each triplet set of (n
x
, n
y
, n
z
) values 
correspond to a point in this space. If n
r
 is a vector 
from origin to point defined by coordinates (n
x
, n
y
, 
n
z
), then the length of vector is expressed as  
) 7 . 6 (
2 2 2
z y x
n n n n + + = 
 
Figure 6.3: The integer space for determination of 
number of modes. 
 
The total number of wavelengths lying between ? and 
?+d ? is same as the number of points in integer space 
whose distance lies between n and n+dn. This is the 
volume of spherical shell of radius n and thickness dn 
i.e. 4 pn
2
dn. Our interest lies in the octant of the shell 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 66 
 
Page 5


 
 
 
 
QUANTUM PHYSICS 
 
 
1. Origin of Quantum Physics. 
2. Wave Particle Duality. 
3. Schrodinger’s Formulation. 
4. Applications of Schrodinger’s Equation. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 64 
 
ORIGIN OF QUANTUM PHYSICS 
 
6.1 Black Body Radiation Distribution 
6.1.1 Features of Black Body Spectrum 
Every substance emits electromagnetic radiations, the 
character of which depends upon the nature and 
temperature of the substance. Some of the important 
features common to all bodies are: (i) When a body is 
placed in surrounding medium which is at a different 
temperature than that of body, then heat exchange 
takes place between the body and medium bilaterally. 
In this process, hot body (or medium) gets cooled 
while other gets hotter. This process continues till 
thermal equilibrium is reached. (ii) The bodies which 
are good emitter in their hot state are good absorbers 
in their cold state. (iii)  A perfect black body is one, 
which absorbs all the incident radiation at low 
temperatures and emits radiation of all the possible 
wavelengths, when heated to a high temperature. 
 
6.1.2 Ferry’s Black Body 
The common example for a practical demonstration of 
black body is the one devised by Ferry (see figure 6.1).  
 
Figure 6.1: The schematic view of Ferry’s Black Body. 
 
It consists of a spherical hollow metallic enclosure 
whose inner surface is coated with lamp black. This 
enclosure has a fine orifice on its surface and a conical 
projection on inner surface diametrically opposite to 
the orifice. The EM radiations entering the orifice, 
strike the projection and further get absorbed due 
multiple reflections through inner surface. In its hot 
state, radiations of all possible wavelengths are 
emitted through the fine orifice. 
 
6.1.3 Black Body Spectrum 
The distribution of energy among various wavelengths 
emitted is a function of temperature only irrespective 
of shape and size of the black body. The spectral 
energy density u(?)d? (the rate of energy density of 
radiation having frequency between ? and ?+d?) are 
plotted as a function of frequency of emitted 
radiation, then following graphical behavior (see 
figure 6.2), called black body radiation spectrum, is 
observed. The following are interesting features 
observed in this radiation energy distribution 
spectrum: 
6.1.3.1: Continuous Energy Distribution 
At each temperature, the black body emits radiations 
of all possible frequencies between minimum and 
maximum limits. This spectral energy distribution is 
continuous and non-uniform. The energy associated 
with a particular frequency increases with rise in 
temperature of the black body. The maximum 
possible frequency emitted by the black body 
increases with temperature of the black body. 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 65 
 
 
Figure 6.1: The black body radiation distribution is 
shown as function of emitted wavelength and also as 
a function of temperature. 
 
6.1.3.2: Wien’s Displacement Law 
The frequency corresponding to which the spectral 
energy density (emitted by the black body) is 
maximum is called frequency of maximum emission. 
The corresponding wavelength, called the wavelength 
of maximum emission, shifts towards smaller values 
when the temperature of the black body is raised. This 
is called Wien’s displacement law. It is mathematically 
expressed as  
 ) 1 . 6 ( b T
m
= ? 
Where b = 2.898x10
-3
mK is called Wien’s constant. 
 
6.1.3.3: Fifth Power Law 
The energy emitted by the black body, corresponding 
to the wavelength of maximum emission, varies as 
fifth power of the absolute temperature of the black 
body. This is called Fifth Power Law. 
 
 
6.1.3.4: Stefan-Boltzmann Law 
The area under the curve represents the energy 
emitted per unit area per second by the black body 
corresponding to all the frequencies emitted by it. 
This area is found to be proportional to the fourth 
power of the absolute temperature of the black body. 
The Stefan’s law states that the energy emitted per 
unit area per second corresponding to all the 
wavelengths emitted by the black body is directly 
proportional to the fourth power of the absolute 
temperature. Mathematically  
 ) 2 . 6 (
4
T E s = 
The above formula gives the energy emitted by the 
black body per unit area per second when it is 
surrounded by black bodies at 0
o
K. Here s=5.67 x 10
-8
 
Wm
-2
K
-4
 is called Stefan’s constant. If the black body is 
surrounded by black bodies at temperature T
o
, then 
this law takes the form given as: 
) 3 . 6 ( ) (
4
0
4
T T E - = s 
If the body is not perfectly black and has emissivity e, 
then  
 ) 4 . 6 ( ) (
4
0
4
T T E - = es 
 
6.1.4 Rayleigh Jeans Treatment 
The shape of the black body radiation distribution 
curve is shown in the figure 6.2. This spectral energy 
distribution is dependent solely on the temperature of 
the black body and does not depend upon shape or 
dimensions of the body. Rayleigh and Jean made a 
theoretical attempt to explain the shape of the curve 
purely on the considerations of classical physics.  
The black body can be simulated as a metallic cavity 
having a fine hole in it and whose inner walls are 
perfect reflectors. The incident radiations, falling on 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 65 
 
this hole, get absorbed following multiple reflections 
inside the cavity. This cavity, when heated, results in 
excitation or even ionization of metal atoms which 
subsequently result in emission of radiation. The 
spectral distribution of radiation, emitted by such a 
body in its hot state, has similar features as that of a 
black body. The emitted radiation suffer multiple 
reflections through walls of the cavity and superpose 
to form standing waves. These standing waves, having 
nodes at inner surface of cavity, oscillate in different 
modes thereby resulting in radiations of all possible 
wavelengths or frequencies. These form so called the 
cavity radiations.  
The electromagnetic radiations inside such a cavity, at 
temperature T, were considered to be three-
dimensional standing waves. The condition for 
standing waves in such a cavity is that the path length 
from wall to wall, in whatever direction, must be 
integral multiple of half wavelengths and hence a 
node occurs at the reflecting surface. If the cavity is 
considered to be a cube of edge L, then standing 
waves must satisfy the following conditions in x, y and 
z directions. 
) 5 . 6 (
,... 3 , 2 , 1
2
,... 3 , 2 , 1
2
,... 3 , 2 , 1
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= =
= =
= =
z z
y y
x x
n
L
n
n
L
n
n
L
n
?
?
?
 
For standing waves in arbitrary direction, we must 
have an independent condition that: 
) 6 . 6 (
2
2
2 2 2
?
?
?
?
?
?
= + +
?
L
n n n
z y x
 
In expression (6.8), the quantity L denotes the path 
length in the arbitrary direction and not the edge 
length of cubical enclosure.  
Let’s calculate the number of standing waves n( ?)d ? 
per unit volume having wavelength between ? and 
?+d ?. This problem is same as determining the 
number of triplet sets (n
x
, n
y
, n
z
) that yield 
wavelengths in the interval ? and ?+d ? and then 
divide this by volume L
3
 of the cavity. 
For this purpose we can imagine an integer space (see 
figure 6.3) whose coordinate axes are n
x
, n
y
 and n
z
 
respectively and each triplet set of (n
x
, n
y
, n
z
) values 
correspond to a point in this space. If n
r
 is a vector 
from origin to point defined by coordinates (n
x
, n
y
, 
n
z
), then the length of vector is expressed as  
) 7 . 6 (
2 2 2
z y x
n n n n + + = 
 
Figure 6.3: The integer space for determination of 
number of modes. 
 
The total number of wavelengths lying between ? and 
?+d ? is same as the number of points in integer space 
whose distance lies between n and n+dn. This is the 
volume of spherical shell of radius n and thickness dn 
i.e. 4 pn
2
dn. Our interest lies in the octant of the shell 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 66 
 
where all the n
x
, n
y
 and n
z
 are positive. Consequently 
the number of points in this octant is N(n)dn given by: 
) 8 . 6 (
2
) 4 (
8
1
) (
2 2
dn n dn n dn n N
p
p = = 
From (6.6) and (6.7), we have  
) 10 . 6 (
2
) 9 . 6 (
2
2
2
2
?
?
?
d
L
dn
L
n
- =
?
?
?
?
?
?
=
 
Since an increase in ? corresponds to decrease in n, 
the total number of permissible wavelengths in the 
cavity is 
) 11 . 6 (
4 2 2
2
) ( ) (
4
3
2
2
?
? p
?
? ?
p
? ?
d L
d
L L
dn n N d N
=
?
?
?
?
?
?
=
- =
 
The number of standing waves per unit volume is 
) 12 . 6 (
4
) (
1
) (
4 3
?
? p
? ? ? ?
d
d N
L
d n = = 
Since there are two perpendicular modes of 
polarization associated with each wavelength so 
equation (6.12) must be multiplied by two to get the 
number of standing waves per unit volume: 
) 13 . 6 (
8
) (
4
?
? p
? ?
d
d n = 
Let’s express the same in terms of frequency. 
) 15 . 6 (
) 14 . 6 (
2
?
?
?
?
?
cd
d
c
- =
=
 
Putting equations (6.15) and (6.16) in (6.13), we get 
) 16 . 6 (
8
) (
3
2
c
d
d n
? p ?
? ? - = 
To obtain the spectral energy density distribution of 
the black body, we must find the average energy 
associated with each standing wave. According to the 
theorem of equipartition of energy, the average 
energy associated with each degree of freedom of a 
system in dynamical equilibrium is kT
2
1
. Since the 
electromagnetic radiations are emitted by oscillators 
which have two degrees of freedom, hence average 
energy associated with each standing wave is kT .  
) 17 . 6 (
8
) (
3
2
c
kTd
d u
? p?
? ? = 
The equation (6.17), referred to as Rayleigh-Jeans 
formula, reveals that rate at which energy is radiated 
increases as ?
2
 and must approach infinity in the 
region of high frequencies. This is not in accordance 
with experimental results depicted in the figure 6.2 
which shows an exponential like fall at higher 
frequencies. This problem encountered in Rayleigh-
Jeans treatment is referred to as Ultraviolet 
Catastrophe.  
 
6.1.5 Planck’s Treatment 
The Rayleigh-Jean treatment could not reproduce the 
experimentally observed spectral energy density 
distribution due to the fact that the theorem of 
equipartition of energy is valid for continuous 
distribution of possible energies. Max Planck 
explained the required distribution using a novel 
assumption that the body emits or absorbs radiation 
in the form of packets called quantum of radiation. 
The energy (E) of the quanta of radiation of frequency 
? is given by ? h E = . The average energy carried by 
quantum of frequency between ? and ?+d ? was 
obtained by assuming Boltzmann distribution as 
follows: 
Dr. J.K.Goswamy’s Lecture Notes: Quantum Physics Page 67