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UNIT-01 
MODERN PHYSICS 
Introduction 
 
 The classical concept of particle, space and time stood unchallenged for 
more than two hundred years and it had achieved many spectacular successes 
particularly in celestial mechanics. But in the early years of the twentieth 
century, the outcome of revolutionary theories like quantum theory and theory 
of relativity swept away the classical concept of particle, space and time given 
by Newton. A new set of laws of quantum physics and relativistic physics 
replaced the laws of classical physics. 
 In classical physics it is assumed that light consists of minute particles 
called corpuscles, which is responsible for various processes and phenomenon 
associated with light; however, after the discovery of phenomenon like 
interference, diffraction and polarization, it is proved beyond doubt that light is 
a form of wave, more correctly electromagnetic wave and these phenomena are 
successfully explained on the basis of Huygens wave theory of light. The 
observed phenomena like Compton Effect and explanation of spectrum of black 
body radiation required description of radiation in terms of particles of energy-
photons. Thus dual nature of light is a fact of experimental evidence. 
Overview of Unit-01 
 
This unit consists of three lessons of teaching. In the first lesson, we will study 
spectrum of black body radiation, significance of Quantum theory. In the 
second lesson, we will study Compton Effect and its significance; in the third 
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lesson, we will study dual nature of radiation and de-Broglie concept of matter 
waves and numericals. 
Objectives of Unit 01 
At the end of this unit we shall understand that: 
 The emission and absorption of energy is not continuous, but 
discrete. 
 A particle in motion is associated with waves called matter waves. 
 Matter has dual characteristics i.e. it exhibits both wave and particle 
properties. 
 Both wave properties and particle properties of moving objects cannot 
appear together at the same time because there is a separable link. 
 De-Broglie waves are pilot waves and not electromagnetic waves. 
 A moving particle is described in terms of wave packet. 
 The dual nature of radiation has made position of a particle 
uncertain. 
Introduction: 
In this unit we will study about the failure of classical physics to explain the 
spectrum of black body radiation leading to discovery of Quantum theory of 
radiation, which signifies the particle nature of radiation thereby opening new 
way of understanding physics. Hence physics developed from the year 1901 is 
called Modern Physics and most of the phenomena are satisfactorily explained 
on the basis of Quantum theory of radiation. Later it became a tool to study 
particles of sub atomic world. And there was a need for new mechanics to 
explain experimentally verified atomic phenomena. 
Objectives: 
At the end of lesson you shall understand that: 
The classical physics cannot explain spectrum of black body radiation, which 
has lead to discovery of Quantum theory of radiation, hence radiation cannot be 
emitted continuously as predicted in classical physics. 
Introduction: 
In this lesson we will study spectrum of black body radiation and various laws 
put forward to explain the energy distribution in the spectrum, their failure and 
success. 
Introduction to Black Body Radiation Spectrum 
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A perfect black body is the one which absorbs the entire radiations incident on 
it, it neither reflects nor transmits radiations, and hence it appears perfectly 
black. But there are no perfect black bodies. For all practical purposes we take 
lamp black as black body, because when a body coated with lamp black 
exposed to radiations, it absorbs 99 percent of it, and also when it is heated, it 
emits radiations containing almost all wavelengths. 
 
The black body radiation is characteristic of its temperature; hence it is 
important to know how the energy is distributed among various wavelengths at 
different temperatures. 
Number of scientists carried out experiments on this energy distribution. 
Among them two scientists namely Lummer and Pringsheim found that when a 
graph of energy density is plotted against wavelength, curves are obtained as 
shown in the figure. These curves are known as radiation curves or spectrum of 
black body radiation. 
The following conclusions can be drawn from the radiation curves. 
1) The energy is not uniformly distributed in the spectrum of black body 
radiation. 
2) At a given temperature, energy density increases with wave length, 
becomes maximum for a particular wavelength and then decreases as 
wavelength increases. 
3) As temperature increases, intense radiation represented by peak of the 
curve shifts towards shorter wavelength region. 
Spectrum of Black Body Radiation or Radiation Curves 
 
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Laws of black body radiation 
In order to explain the spectrum of black body radiation, number of laws have 
been put forward, notable among them are Stefan‘s law of Radiation, Rayleigh-
Jeans Law of energy distribution, Wien‘s Law of energy distribution and 
Planck‘s Law of Radiation. 
Stefan’s law of radiation 
The Stefan‘s law states that energy radiated per second per unit area is directly 
proportional to thefourth power of absolute temperature. E  T 4, or E =  T4 
where is Stefan‘s constant, though this law is experimentally verified, it does 
not explain the energy distribution in the spectrum of black body radiation. 
 
Wien’s law of radiation 
In the year 1893, Wien assumed that black body radiation in a cavity is 
supposed to be emitted by resonators of molecular dimensions having 
Maxwellian velocity distribution and applied law of kinetic theory of gases to 
obtain formula for energy distribution as UdC1
e–(C2/T)d, where Ud is 
the energy /unit volume for wavelengths in the range,  and dand C1 and 
C2 are constants. 
 
Drawbacks of Wien’s Law: This law explains the energy distribution only in 
shorter wavelengths & fails to explain the energy distribution in longer 
wavelength region. Also according to this law, when temperature is zero, energy 
density is finite. This is a contradiction to Stefan‘s law. 
 
Lord Rayleigh –Jeans law of Radiation: 
Lord Rayleigh–Jeans considered the black body radiations full of 
electromagnetic waves of all wavelengths, between 0 and infinity, which due to 
reflection, form standing waves. They calculated number of possible waves 
having wavelengths between and +d and by using law of equi-partition of 
energy, they established distribution law as: Ud= 8kT
-4dBecause of the 
presence of the factor -4in the equation, the energy radiated by the black body 
should rapidly decrease with increasing wavelength. 
 
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Drawbacks of Rayleigh–Jeans law: It is found that, Lord Rayleigh–Jeans law 
holds good only for longer wavelengths region and fails to explain energy 
distribution in shorter wavelength region, moreover; as per this law, as 
wavelength decreases, energy density increases enormously deviating from the 
experimental observations. The failure of the Rayleigh–Jeans law to explain the 
aspect of very little emission of radiation beyond the violet region towards the 
lower wavelength side of the spectrum is particularly referred to as Ultra-violet 
Catastrophe. 
 
Planck’s Law of Radiation 
In the year 1901, Max Planck of Germany put forward Quantum Theory of 
Radiation to explain Black Body Radiation spectrum. The following are the 
assumptions of Planck law of radiation. 
1) The black body radiations in a cavity are composed of tiny oscillators 
having molecular dimensions, which can vibrate with all possible 
frequencies.2) The frequency of radiations emitted by oscillators is same as the 
frequency of its vibrations. 
3) An oscillator cannot emit energy in a continuous manner, but emission 
and absorption can take place only in terms of small packet of energy 
called Quanta, the oscillator can have only discrete energy values E given 
by nh ν ν= Frequency of radiations, n = integer and ‗h‘ is Planck‘s constant, h= 6.625 x 
10-34Js. 
Planck using above assumptions derived a formula to explain black body 
radiation spectrum as, 
 [Since, ν=c/λ] ----------- (1) 
This is called Planck’s radiation law and explains the entire spectrum of 
black body radiation. From this law, we can also obtain Stefan‘s law, Wien‘s 
law and Rayleigh-Jean law under suitable conditions. 
1. Reduction of Planck’s radiation law to Wien’s law for shorter 
wavelengths: 
For shorter wavelengths, ν=c/λ is large, 
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When ν is large, is very large 
>>1. 
( -1)≈ 
Making use of this in (1) 
, 
=C1 -5 where C1=8πhc and C2=(hc/k). 
This equation is Wien’s law of radiation. 
2. Reduction of Planck’s radiation law to Rayleigh-Jeans law for longer 
wavelengths: 
For longer wavelengths, ν= c/λ is small, 
When ν is small, hν/kt will be very small. 
Expanding as power series, we have, 
=1+ (hν/kt) + (hν/kt)2+….. 
 1+hν/kt [since hν/kt is very small, its higher power terms could be 
neglected] 
( -1) hν/kt = hc/λkt 
Substituting in (1) 
 
 
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This equation is Rayleigh-Jeans law of radiation. 
Thus Wien‘s law and Rayleigh-Jeans law are special cases of Planck‘s law. 
Rayleigh-Jeans Law
Planck’s law
Wien’s law

Ed
Energy distribution curves
 
Summary of Lesson –01 
Here we have learnt that classical physics cannot explain black body radiation 
spectrum. The emission and absorption of energy takes place only in terms of quanta 
and not continuously as predicted in classical physics. Quantum theory of radiation 
has opened a new concept of understanding physics. 
LESSON-2 
Objectives: 
At the end of lesson you shall understand that: 
Light rays consists of invisible particles called photons. 
A single electron in metal cannot absorb one photon of energy h. 
Compton scattering is different from classical scattering. 
Compton effect signifies particle nature of radiation. 
Introduction 
In this lesson, we will study Compton Effect which signifies particle nature of 
radiation, thereby strengthening the fact that radiation has dual 
characteristics. 
COMPTON EFFECT 
In the year 1924, Compton discovered that when monochromatic beam of very 
high frequency radiation such as X-rays or Gamma rays is made to scatter 
through a substance, the scattered radiation found to contain two components; 
one having same frequency or wavelength as that of incident radiation, known 
as unmodified radiation; and the other, having lower frequency or longer 
wavelength than incident radiation known as modified radiation. This is called 
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Compton scattering, during the process an electron recoils with certain velocity. 
This phenomenon is called Compton Effect. 
The Compton Effect is explained on the basis of Quantum theory of radiation, 
in which it is assumed that, radiation is composed of small packets of energy 
called Quanta or photons having energy h. According to Compton, when a 
photon of energy hof momentum h/ moving with velocity equal to velocity of 
light, obeying laws of conservation of energy and momentum, strikes an 
electron which is at rest, there occurs an elastic collision between two particles 
namely photon and electron. 
When photon of energy h strikes the electron at rest, photon transfers some of 
its energy to electron, therefore photon loses its energy, hence, its frequency 
reduces to 1 and wavelength changes to , the scattered photon makes an 
angle  with the incident direction, during the process an electron gains kinetic 
energy and recoils with certain velocity. 
 
Compton by applying laws of conservation of energy and momentum showed 
that, the change in wavelength is given by formula, 
 
 
 
where mo = rest mass of electron. 
The change in wavelength 
 '
 is called Compton shift. This shows that the 
change in wavelength (Compton shift) depends neither on the incident 
wavelength nor the scattering material, but depends only on the angle of 
scattering. 
Experimental Arrangement to Study Compton Effect 
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cm
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
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The Experimental arrangement to study Compton Effect is, as shown in the 
figure, a monochromatic beam of very high frequency such as X-rays of known 
wavelength is made to fall on a scattering substance such as graphite. The 
intensity of scattered X-rays for different angles of scattering is measured by 
Bragg x-ray spectrometer, and then a graph of intensity versus angle of 
scattering is plotted. When the angle of scattering is 90o, the Compton shift is 
found to be 0.0243 Å. This value is in agreement with theoretical value obtained 
from Compton formula. 
Physical significance of Compton Effect 
The phenomena of Compton effect is explained by Compton on the basis of 
Quantum theory of radiation, in which it is assumed that radiation is composed 
of small packets of energy called Quanta. The Compton Effect is an elastic 
collision between two particles namely photon and electron in which exchange 
of energy takes place as if it is a particle–particle collision. Also it is assumed 
that photon and electron obey laws of conservation of energy and momentum. 
Hence Compton Effect signifies particle nature of radiation. 
Summary of Lesson -02 
The Compton Effect signifies particle nature of radiation. 
LESSON-3 
Objectives 
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 At the end of this lesson we will learn that: 
Any particle in motion exhibits wave like properties. 
Matter waves are generated due to motion of the particle not by the 
charge carried by them. 
Wave properties and particle properties do not appear together. 
Dual nature of radiation has put the position of the particle uncertain. 
Introduction: 
In this lesson we shall study dual characteristics of matter waves and de-
Broglie concept that all particles in motion exhibit wave properties and also de-
Broglie equation. 
Wave Particle Dualism and de-Broglie concept of Matter Waves 
Before we discuss wave particle dualism, we must know the concept of particle 
and the concept of wave. The concept of particle is easy to understand, because 
it has mass and occupies certain fixed position in space and particle in motion 
has definite momentum; when slowed down, it gives out energy. Therefore 
particle is specified by its mass, momentum, energy and position. 
The concept of wave is bit difficult to understand, because a wave is a 
disturbance spread over a large area. We cannot say wave is coming from here 
or going there. No mass is associated with wave and the wave is characterized 
by its wavelength, frequency, amplitude and phase. 
Considering the above properties of particle and wave, it is difficult to accept 
the dual nature of radiation, but the acceptance is necessary because, the 
phenomenon like interference and diffraction has shown beyond doubt the wave 
nature of light radiation. And successfully explained by Huygens wave theory of 
light, however, experimental phenomenon like Photo electric effect, Compton 
Effect are successfully explained by Quantum theory of radiation, which 
signifies particle nature of radiation. Hence we can conclude that radiation has 
dual characteristics i.e. sometimes behaving like a wave and at other time as 
a particle, but radiation cannot exhibit both wave and particle properties 
simultaneously.De-Broglie concept of matter waves 
L. de-Broglie in the year 1924 put forward the concept of matter waves. 
According to this concept the dual characteristics of radiation is not confined 
only to electromagnetic waves, but also holds good for all material particles in 
motion i.e. all the particles like electrons, protons, neutrons, molecules, atoms 
etc. exhibit dual characteristics. His theory is based on the fact that nature 
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loves symmetry that means when waves exhibits particle like properties then 
particle also should possess wave like properties. 
According to de-Broglie the particle in motion is associated with a group of 
waves and controlled by the wave. This wave is known as matter wave or de 
Broglie wave and wavelength associated with it is called de Broglie wavelength. 
 
 
De-Broglie wavelength of a free particle 
For a free particle, total energy is same as its kinetic energy given by, 
 E = ½ mv2 
 E = m2v2/2m (But p = mv) 
 E= p2/2m 
Hence, 
 p = √2mE 
By de Broglie hypothesis, 
 λ = h/p 
Therefore, λ= h/√2mE = h/√2meV (since E = eV) where V is the 
accelerating potential on an electron. 
Substituting the constants, we get, λ = 12.27/√V Ǻ. 
Characteristics of Matter Waves 
• Matter waves are the waves associated with a moving particle. 
• The lighter the particle larger the wavelength. 
• Smaller the velocity of particle larger the wavelength. 
• The amplitude of the matter wave at a given point determines the 
probability of finding the particle at that point at a given instant of time. 
• The wavelength of a particle is given by, λ= h/p = h/mv 
Summary of Lesson 
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The material particle in motion exhibits wave like properties. The de - 
Broglie waves are pilot waves and are not electro-magnetic waves. Wave 
properties and particle properties cannot appear together.The dual nature of 
radiation has put position of particle uncertain. 
Solved Examples 
1. Calculate the momentum of the particle and de Broglie wave length 
associated with an electron with a KE of 1.5KeV. 
Solution: Data p=? 
λ = ? 
K E =1.5x10³ eV 
 p² = 2mE 
= 2x9.1x10‐³¹x1.5x10³x1.6x10‐¹⁹ 
= 2.08x10-²³ kgms-1 
λ = h/p 
 = 6.625x10‐³⁴/2.08x10-²³ 
 = 3.10x10‐¹¹ m. 
2. Calculate the wave length of the wave associated with an electron of 1eV. 
Solution: 
λ= h/p 
 = h/{2mE}½ 
 = 6.625x10-³⁴/{2x9.1x10-31x1.6x10-¹⁹}½ 
 = 1.23x10-⁹ m 
3. Find de Broglie wave length associated with a proton having velocity equal to 
1/30th of that light. Given, mass of proton as1.67x10-²⁷kg. 
Solution: v =1x3x10⁸/30 = 10⁷ m/s 
λ =h/mv 
 = 6.625x10-³⁴/1.67x10-²⁷x10 ⁷ 
= 3.9x10-¹⁴ m/s 
4. The velocity of an electron of a hydrogen atom in the ground state is 
2.19x10⁶m/s. Calculate the wave length of the deBroglie waves associated with 
motion. 
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Solution: λ = h/mv 
= 6.625x10-³⁴ /9.11x10-31x2.19x10⁶ 
= 3.31x10-¹⁰ m 
 5. Estimate the potential difference through which a proton is needed to be 
accelerated so that its deBroglie wave length becomes equal to 1Å. Given that 
it‘s mass is 1.673x10-²⁷kg. 
Solution: eV = 1/2 mv² 
 = p²/2m 
 = h²/2mλ²{v² = h²/m²λ²} 
 = h²/2meλ² 
 = {6.625x10-³⁴}²/2x1.67x10-27x1.6x10-19x(10-¹⁰)² 
 = 0.082 V 
6. Compare the energy of a photon with that of a neutron when both are 
associated with wave length of 1 Å. Given the mass of the neutron is 1.67x10-
²⁷Kg. 
Solution: E₁ = hν 
= hc/λ₁ 
= 1.989x10-¹⁵ / 10-10x 1.6x10-¹⁹eV 
=12411 eV 
E₂ = h²/2mλ₂² 
 = 0.08 eV 
 E₁/E₂ =12411/0.08 
 = 1.5 x10⁵ 
7. Find the KE of an electron with de Broglie wave length of 0.2nm. 
Solution: p = h/λ 
 = 6.625x10‐³⁴/0.2x10‐⁹ 
= 3.313x10‐²⁴ n-s 
 E = p²/2m 
 = (3.313x10‐²⁴)²/2x9.1x10‐³¹ = 37.69 eV 
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QUANTUM MECHANICS 
Over view 
This unit consists of five lessons, in first lesson we will study Heisenberg‘s 
uncertainty principle and its physical significance. In lesson two, we will study 
the applications of uncertainty principle and show that it is not possible for an 
electron to exist inside the nucleus. In lesson three, we will study wave 
function, its properties and physical significance and also we will study 
probability density and normalized wave functions, Eigen values, Eigen 
functions. In lesson four we will study Schrödinger matter wave equation and in 
the last lesson we will study particle in a box, energy values and wave 
functions. 
Objectives 
At the end of unit we would understand that: 
 In sub atomic world, it is impossible to determine precise values of 
two physical variables of particular pair which describes atomic 
system. 
 Both wave properties and particle properties are essential to get clear 
picture of atomic system. 
 Wave properties and particle properties are complimentary to one 
another. 
 In our daily life, we cannot realize quantum conditions. 
 Particle in a box is a quantum mechanical problem and the probable 
position of a particle can be estimated by evaluating the value of 
│ψ│2. 
 Quantum mechanics is an important tool to study atomic and sub 
atomic state. 
LESSON –1 
Introduction 
In this lesson we will study uncertainty principle, its related equations derived 
from concept of wave packet and also we will study the physical significance of 
uncertainty principle. 
Objectives 
At end of lesson we understand that: 
It is impossible to determine precise values of physical variables which 
describes atomic system. Hence we should always think of probabilities 
of estimating those values. 
Both wave properties and particle properties of moving objects cannot 
appear together at the same time. 
Wave properties and particle properties of moving objects are 
complimentary to one another. 
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From uncertainty principle it is clear that, inaccuracy inherently present 
in its measurements. 
HEISENBERG’S UNCERTAINITY PRINCIPLE 
In the year 1927, Heisenberg proposed very interesting principle known as 
uncertainty principle, which is a direct consequence of dual nature of matter. 
In the classical physics the moving particle has fixed position in space and 
definite momentum. If the initial values are known final values can be 
determined. However in Quantum Mechanics the moving particle is described 
by a wave packet. The particles should be inside wave packet, hence when wave 
packet is small; position of the particle may be fixed, but particle flies off rapidly 
due to very high velocity; hence, its momentum cannot be determined 
accurately. When the wave packet is large, velocity or momentum may be 
determined but position of particle becomes uncertain. 
In this way, certainty in position involves uncertainty in momentum and 
certainty in momentum involves uncertainty in position. Therefore, it is 
impossible to say where exactly the particle inside the wave packet is and what 
its exact momentum is. 
According to uncertainty principle it is impossible to determine precisely and 
simultaneously, the exact values of both members of particular pair of physical 
variables which describes atomic system. 
In any simultaneous determination of position and momentum of a particle, the 
product of corresponding uncertainties inherently present in the measurements 
is equal to or greater than h/4π 
Δp.Δx ≥h/4π 
These are the other uncertainty relations: 
ΔE.Δt ≥ h/4π 
ΔL.Δθ≥ h/4π 
Δx = uncertainty in measurement of position 
Δp = uncertainty in measurement of momentum 
ΔE = uncertainty in measurement of energy 
Δt = uncertainty in measurement of time 
ΔL = uncertaintyin measurement of angular momentum 
Δθ = uncertainty in measurement of angular distance 
Note: Heisenberg‘s uncertainty principle could also be expressed in terms of 
uncertainty involved in the measurements of physical variable pair like angular 
displacement (θ) and angular momentum (L). 
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Summary 
It is impossible to determine the values of both members of a particular pair of 
physical variables which describes atomic system. Hence we should always 
think of probability to estimate those values. However precise may be the 
method of measurement there is no escape from these uncertainties because it 
is an inherent limitation of nature on the measurement. 
LESSON-2 
Objectives: 
At the end of lesson we shall understand that the electron cannot exist inside 
the nucleus of an atom and we can determine frequency of radiation emitted by 
atom and radius of electronic orbit and binding energy of electron. 
Introduction 
In this lesson we will study applications of uncertainty principle, mainly to 
show that it is not possible for an electron to stay inside the nucleus of an 
atom. 
Applications of Heisenberg’s Uncertainty Principle 
 Non-existence of electrons in nucleus of atoms 
 Calculation of frequency of radiation emitted by atom 
 Calculation of binding energy of an electron in an atom 
 Determination of radius of Bohr electronic orbit 
Here we will discuss first two important applications. 
Non-existence of electrons in nucleus of atoms 
The diameter of nucleus of atom is of the order 10-14 m. If an electron exists in 
nucleus of atom then maximum uncertainty in determining position of electron 
must be 10-14 m. 
 ( x) max =10- 14 m 
From uncertainty principle (x) max (p) min = h/4 
 10- 14 (p) min = h/4 
 
If an electron exists in nucleus then it should possess minimum momentum of 
0.528 x 10-20 kg-m/sec particle of having this momentum must be moving 
with velocity equal to velocity of light, then it must be a relativistic problem. 
Hence energy of the particle is given by E = mc2 or E= (mc) (c) 
E = p c = (0.528 x 10-20 kg-m/sec) (3 x 108m ) J 
 E = 0.990346 x 107 eV 
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 E = 10 MeV 
If an electron exists in nucleus of atom then it should have minimum energy of 
10 MeV, but beta decay experiments has shown that energy possessed by beta 
particle from nucleus of an atom has maximum energy of 2 to 3 MeV. Hence we 
can conclude that it is not possible for an electron to exist inside the nucleus of 
an atom. 
Physical significance of Heisenberg’s uncertainty principle 
 Non-existence of electrons inside the nucleus of atoms. 
 Calculation of frequency of radiation emitted by an atom. 
 Calculation of binding energy of an electron in an atom. 
 Determination of radius of Bohr electronic orbit. 
 The wave and particle properties are complimentary to one another. 
 It is impossible to determine precisely and simultaneously values of 
physical variables which describes the atomic system. 
Summary 
The negatively charged particle electron cannot exist inside the nucleus. The 
wave and particle properties are complimentary to one another rather than 
contradictory. 
LESSON-3 
Objectives 
At end of lesson we shall understand that: 
 The wave function by itself has no physical significance 
 The wave function is a complex quantity 
 The value of ││2 evaluated at a point gives the probability of 
finding the particle at that point 
Introduction 
In this lesson we will study the wave function and its characteristics, physical 
significance, probability densities, and normalization of a wave function. 
Wave Function, Probability Density and Normalized Functions 
The concept of wave function was introduced by Schrödinger in the matter wave 
equation. It is denoted by , it is a variable whose variations constitutes matter 
wave. Wave Function is related to position of particle. The following are some 
characteristics of wave function. 
1) The wave function by itself has no direct physical significance. 
2) The wave function cannot be interpreted by an experiment. 
3) The wave function is complex quantity consisting of both real and 
imaginary parts. 
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4) With the knowledge of the wave function we can establish angular 
momentum, energy and position of particle. 
5) The value of ││2 evaluated at point gives the probability of finding 
particle at that point. 
Properties of Wave function: 
Property 1:is single valued everywhere. 
 
 
MULTIVALUED FUNCTION 
A function f(x) which is not single valued over a certain interval as shown in the 
above figure, has 3 values f1,f2,f3 for the same value of P at x=P. Since f1≠f2≠f3, 
it says that the probabilities of finding the particle have 3 different values at the 
same location. Hence such wave functions are not acceptable. 
Property 2:is finite everywhere. 
 
FUNCTION NOT FINITE AT A POINT 
A function f(x) which is not finite at x=R as shown the above figure. At x=R, 
f(x)=infinity. Thus if f(x) were to be a wave function, it signifies large probability 
of finding the particle at a single location at x=R, which violates the uncertainty 
principle. Hence such wave functions are not acceptable. 
 
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Property 3:and its first derivatives with respect to its variable are 
continuous everywhere. 
 
Discontinuous function 
A function which is discontinuous at Q as shown in the above figure, at x=Q, 
f(x) is truncated at A and restarts at B. Between A and B it is not defined and 
f(x) at Q cannot be ascertained. Hence such wave functions are not acceptable. 
Property 4: For bound states,  must vanish at infinity. If is a complex 
function, then must vanish at infinity. 
The wave functions that possess these four properties are named in quantum 
mechanics as Eigen functions. 
Probability Density 
 
The wave function is a complex quantity consisting of both real and 
imaginary parts. Hence it can be expressed as follows: 
ψ = a + ib where a and b are real functions of (x, y, z) and ‗t‘. 
Complex conjugate of ψ is, 
 ψ* = a – ib 
The product of ψ and ψ* is ψψ* = a 2 + b2, which is called 
Probability density denoted by P = ׀ψ׀2, 
Where ψ and ψ* are real and positive and also if ψ ≠ 0. 
Normalized functions 
The value of ׀ψ׀2 evaluated at point gives the probability of finding a 
particle at that point, hence the probability of finding the particle in 
an element of volume δv is given by: 
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 ׀ψ׀2δv 
Since the particle must be somewhere in space, the total 
probability of finding the particle should be equal to 1 i.e 
 ׀ψ׀2δv = 1 
Any function which obeys this condition is said to be normalized 
Wave function. Normalized wave functions should satisfy following 
conditions: 
1. It should be single valued function. 
2. It should be finite everywhere. 
3. It should be continuous and it should have continuous first derivative 
ψ tends to zero when x, y, z tends to 0. 
Eigen functions and Eigen values of energy 
In Quantum mechanics, the state of a system is defined by its energy, position 
and momentum. These quantities can be obtained with the knowledge of wave 
function ψ. 
Hence to define the state of a system we have to solve Schrödinger wave 
equation, but Schrödinger equation is a second order equation. It has several 
solutions, and only few of them are acceptable which gives physical meaning, 
these acceptable solutions are called proper functions or Eigen functions. 
 These are single valued, finite and continuous functions. 
Eigen functions are used in Schrödinger equation to solve for energy of a 
system,since there can only be certain restricted Eigen functions and hence 
only few restricted values of energy, these values of energy is called Eigen 
values of energy. 
Summary 
The wave function is a variable quantity, whose variations constitute matter 
waves. 
The wave function is related to position of particle. 
With the knowledge of wave function we can establish energy, angular 
momentum and position of particle. 
Lesson 4 
Objectives 
At the end of the lesson we will understand that: 
21 
 
The Schrödinger Wave Equation is useful in obtaining wave function, which is 
related to position of a particle. We will also understand that energy of free 
particle is not quantized. 
Introduction 
In this lesson we study, which is fundamental equation of quantum mechanics 
and free particle. 
Schrödinger Time Independent Wave Equation 
According to de Broglie concept of matter waves, a particle in motion is 
associated with group of waves called matter waves, the wavelength is given by 
λ = h/mv. 
If the particle behaves like a wave then there should be some sort of wave 
equation which describe behavior of wave, and this equation is called 
Schrödinger Time Independent Wave Equation. 
Consider a system of stationary waves, a particle of coordinates (x,y,z,) and 
wave function ψ. The wave equation of wave motion in positive x – direction is 
given by, 
 Ψ = Aei(kx–ωt) = …………………. 1 
The time independent part is given by, 
 ψ = Aeikx…………………..2 
 Ψ = ψe–iωt…………………3 
Let us differentiate Ψ twice with respect to x, 
We get, 
 ∂2 Ψ = e –iωt ∂2 ψ ………..4 
 ∂x2 ∂x2 
 
Let us differentiate Ψ twice with respect to t, 
 ∂2Ψ = - ω2 e –iωtψ ………5 
 ∂t2 
 
We have the equation for a travelling as, 
 d2y = 1 d2 y 
 dx2 v2 dt2 
where y is the displacement and v is the velocity of the wave. 
By analogy, we can write the motion of a free particle as, 
 d2Ψ = 1 d2 ψ ……………6 
 dx2 v2 dt2 
 
22 
 
The above equation represents waves propagating along x–axis with a velocity v 
and Ψ is the displacement at the instant t. Substituting 4 and 5 in 6 we get, 
d2Ψ = - ω2 ψ ……………7 
dx2 v2 
 
d2Ψ = - 4π2 ψ 
dx2 λ2 
 
or 1/ λ2 = - 1 d2 ψ ……………8 
 4π2ψ dx2 
 
We have KE = p2/2m 
Put λ = h/p, KE = h2/2m. 1/λ2 ………….9 
Substituting 7 in 9 we get, 
KE = -h2/8π2m. 1/ ψ .d2 ψ /dx2……………10 
E = KE + PE = -h2/8π2m. 1/ ψ .d2 ψ /dx2 + V 
E – V = -h2/8π2m 1/ ψ d 2 ψ /dx2 
d2 ψ /dx2 + 8π2m/h2 ( E – V) ψ = 0 
This is Schrödinger time independent wave equation. 
 
Summary 
The Schrödinger matter wave equation is basic equation of quantum 
mechanics. And it is one of the important tools to study subatomic world. The 
energy of free particle is not quantized. 
Lesson 5 
Objectives 
At the end of the lesson we shall learn that: 
 The energy levels for a particle in a box are quantized and hence cannot 
have arbitrary values. 
 The energy corresponding to n=1 is called ground state energy or zero 
point energy and all other energy states are called excited states. 
 The energy difference between successive levels is quite large. 
 The electron cannot jump from one level to the other level on the 
strength of thermal energy, hence quantization of energy plays important 
role in case of electron. 
Introduction 
23 
 
In this lesson we will study one of the important applications of Schrödinger 
wave equation, that is particle in a box of infinite depth and solve for Eigen 
values and Eigen functions. We will also study wave functions, probability 
densities and energy values of a particle in a box. 
Particle in a box of infinite depth 
Considered a particle of mass‘ ‗m‘ moving along ‗x‘ axis between two rigid walls 
of infinite length at x=0 and x= a. The particle is said to be moving inside 
potential well of infinite depth and potential inside box is zero and rises to 
infinity outside box. 
i.e V=0 for 0 ≤ x ≤ a, V= ∞ for 0 ≥ x x ≥ a 
 
 
 
 
 
 
Schrödinger wave equation for particle in a box 
δ2ψ + 8π2m (E — V) ψ = 0 
δx2 h2 
 
but V=0 for 0 ≤ x ≤ a 
δ2ψ + 8π2m( E ) ψ = 0 
δx2 h2 
 
or E ψ = - h2/ 8π2m . (δ2ψ/ δx2) 
put, 8π2mE = K2 
 h2 
δ2ψ + K2ψ = 0 
δx2 
 
The general solution for above equation is of the type, 
Ψ(x) = A sinKx + B cosKx 
‗A‘ and ‗B‘ are constants to be determined by applying suitable boundary 
conditions. 
The particle cannot exist outside the box and cannot penetrate through walls, 
hence the wave function Ψ(x) must be zero for x=o and x= a. 
24 
 
i) Applying boundary conditions Ψ(x)=0 at x = 0, 
We get 0 = A sin K(0) + B cosK( 0) 
 0 = A (0) + B (1) 
 ie B = 0, 
Substituting in above equation, Ψ (x) = A sin Kx 
 ii) Applying boundary conditions Ψ(x) = 0 at x = a, 
 0 = A sin Ka 
Since, A ≠ 0, sin Ka = 0 
Hence ka = 0, π, 2π, 3π, 4π, 5π, ------nπ 
Ka = nπ 
or K = nπ/a. 
The wave function is Ψ(x) = A sin(nπ)x for n = 1,2,3,4,5,6,7,8---- 
 a 
 
Energy 8π2 m E = K2 
 h2 
 
Hence, 8π2mE = n2π2 
 h2 a2 
 
E=n2h2 
 8ma2 
From the above equation it is clear that particle in a box cannot have arbitrary 
value for its energy, but it can take values corresponding to n = 1,2,3,4, --- 
these values are called Eigen values of energy. 
 
EIGEN VALUES FOR ENERGY 
When n = 1, E1 = h2 = Eo-is called ground state energy 
 8ma2 
Eo or zero point energy or lowest permitted energy. 
When n = 2, E2= 4h2 = 4Eo - this is first excited energy state 
 2ma2 
When n = 3, E3= 9h2 = 9Eo - this is second excited energy state 
 8ma2 
To evaluate A in Ψ (x) = A sin(nπ/a)x, We have to perform normalization of 
wave function. As the particle is inside the box at any time, we can write, 
a 
25 
 
∫ ׀ψ׀2δx = 1 
0 
 
Substituting for ψ [from Ψ(x) = A sin (nπ/a)] 
We get, a 
 ∫ A2sin2 (nπ/a)x δx = 1 
 0 
Solving we get, A = √2/a 
Substituting in the equation for Ψ(x) = A sin(nπ/a)x 
We get, 
Ψ(x) = √2/a sin(nπ/a)x 
ENERGY EIGEN FUNCTIONS 
For n=1, 
Ψ1 = √2/a sin (π/a) x. Here Ψ1= 0 for x = 0 & x = a. And maximum for x = 
a/2. 
For n=2, 
Ψ2 = √2/a sin (2π/a)x. Here Ψ2= 0 for x=0, a/2 and a. 
And maximum for x=a/4 and 3a/4. 
For n=3, 
Ψ3 = √2/a sin(3π/a)x. Here Ψ3= 0 for x=0, a/3, 2 a/3 and a. 
And maximum for x=a/6, a/2 and 5a/6. The plot of ׀ψ(x)׀2 versus x is as shown 
in the figure below, for n=1,2 and 3. 
 
 
 
 
 
 
Summary: 
The particle in a box is quantum mechanical problem. The most probable 
position of a particle at different energy levels can be estimated by solving for its 
Eigen functions. The existence of zero point energy is in conformity with 
Heisenberg uncertainty principle. 
26 
 
 
Solved problems 
1. An electron has a speed of 300m/s accurate to 0.01% with what 
fundamental accuracy can we locate the position of the electron? 
Solution: Given, v= 300m/s, ∆v=0.01% of v. 
 ∆v = (0.01/100) x 300=0.03m/s. 
We know that uncertainty relation is ∆x.∆p≥ h/4π 
For the given uncertainty in speed, ∆p is minimum 
Uncertaintyin the position is given by, 
∆x= h/4π∆p= h/4πm∆v 
= 6.632x10-34 / 4x 3.14x 9.11x10-31x3x10-2 
=1.93x10-3m/s 
2. An electron of energy 20 eV is passed through a circular hole of radius10-6 m. 
what is the uncertainty introduced in the angle of emergence? 
Solution: Given E= 250eV= 250x1.6x10-19J 
r=10-6m 
∆x=2x10-6 
We know that energy E=p2/2m and 
p=√2mE 
=√(2x9.11x10-31x40x10-18)= 8.853x10-24Kgm/s. 
∆p= h/4π∆x 
=6.632x10-34 / 4x 3.14x 2x10-6 = 0.263x10-28Kgm/s 
Angle of emergence, =∆p/p 
=0.263x10-28 /8.853x10-24 
=0.0309x10-4radian 
3.The average time an atom retains excess excitation energy before re-emitting 
it in the form of electromagnetic radiation is 10-8 sec. calculate the limit of 
accuracy with which the excitation energy of the emitted radiation can be 
determined? 
Solution: Given ∆t=10-8 sec 
27 
 
According to uncertainty principle, ∆E.∆t= h/4π 
∆E = h/4π∆t 
=6.632x10-34 / 4x 3.14x 10-8 
= 0.5x10-26 J 
=0.5x10-26/1.6x10-19eV 
∆E=0.3x10-7eV. 
4. Using Heisenberg uncertainty relation, calculate the kinetic energy of an 
electron in a hydrogen atom? 
Solution: The uncertainty in the coordinate of an electron inside the atom is 
equal to the radius of the atom. The Bohr radius, r= 0.053nm,is the reasonable 
estimate for the uncertainty in position ∆p. 
We know from Heisenberg uncertainty principle. 
∆x. ∆p=h/4π 
∆p= h/4π∆x 
But ∆x=r (Bohr radius) 
Now energy, E=p2/2m= h2/16π2r2 2m 
=(6.632x10-34)2/ (4x 3.14x0.053x 10-9)2x2x9.1x10-31 
=5.45x10-19 J = 3.4 eV 
5. An electron is constrained in a 1-dimensional box of side 1nm. Calculate 
the first Eigen values in electron volt. 
Solution: Given a=1nm 
The Eigen values are given by 
En=n2h2/8ma2 
The first Eigen value given by 
E1= (6.632x10-34)2 / 8x9.11x10-31x (1x 10-9)2J 
=0.377eV 
E1=0.377eV 
Second Eigen value, E2= 22x 0.377= 1.508 eV 
Third Eigen value, E2= 32x 0.377= 3.393 eV 
28 
 
Fourth Eigen value, E2= 42x 0.377= 6.5032 eV 
6. Is it possible to observe energy states for a ball of mass 10 grams moving 
in a box of length 10cm. 
Solution: The Energy is given by 
En=n2h2/8ma2 
E1= n2(6.632x10-34)2 / 8x9.11x10-31x (0.1)2 
=38x10-18 n2 
When n=1, 2 , 3etc energies are 38x10-18 eV, 152.6x10-18eV, 
343.35 x 10-18eV. 
The energies states are so near that they appear as continuous. 
7.A spectral line of wavelength 5461 Å has a width of 10-4 Å. Evaluate the 
minimum time spent by the electrons in the upper energy state between the 
excitation and de-excitation processes? 
Solution: Wavelength of the spectral line, λ=5461x10-10m 
Width of the spectral line ∆ λ=10-14m 
Minimum time spent by the electron, ∆t=? 
We have the equation, Е = hν= hc/ λ 
∆Е =hc ∆(1/λ) 
∆Е =hc (∆λ/λ2) ------ (1) 
As per uncertainty principle, 
∆E ∆t ≥ h/4π 
∆t ≥ h/4π∆E ------ (2) 
From (1) the right hand side of (2) can be written as 
h/4π∆E = h λ2/4π (hc ∆λ)= λ2/4πc∆λ 
= (5461 x 10-10)2/4π x 3 x108 x10-14 
=0.8X10-8s 
From equation (2), we have ∆t = 0.8X10-8 second. 
Therefore the minimum time spent by the electron is ≥ 0.8X10-8 second. 
Xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 
 
 
29 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
UNIT-II 
LESSON I 
Objectives 
At the end of this lesson we shall understand that valence electrons in metal 
are treated as molecules of gas. The free electrons in metal move from atom to 
atom among positive ion cores. The flow of current in metal is the consequence 
of drift velocity. 
Introduction 
In this lesson, we will study about free electron theory proposed by Drude to 
explain some of the outstanding properties of metals and drift velocity and its 
significance. 
30 
 
ELECTRICAL CONDUCTIVITY 
3.1 Drude – Lorentz’s Free Electron Theory of metals: [Classical Free 
Electron Theory] 
In order to explain some of the outstanding properties of metals like very high 
electrical, thermal conductivities in the year 1900; Drude proposed free electron 
theory of metals. According to this theory valence electrons in metals are free to 
move inside the metal among positive ionic cores, these electrons are called 
conduction electrons and they are responsible for most of the properties of 
metals. 
It is assumed that the metal consists of large number of atoms, which are held 
together, thus valence electrons move from atom to atom throughout the metal. 
When an electron leaves the parent atom, the atom losses its electrical 
neutrality, thus it is ionized. These ions are localized in the metal and the 
structural formation of these ions in three dimensional arrays is known as 
lattice. The positive ions are fixed in the lattice frame but constantly vibrating 
about their mean positions due to thermal agitations. The thermal agitation is 
responsible for random motion of electrons in the metal, hence free electrons in 
metal is treated as molecules of gas called electron gas. 
 
When an electric field is applied, the negatively charged electrons moves along 
the direction of positive field and hence produce current in the metal. In order 
to prevent the electrons accelerating indefinitely it is assumed that they collide 
elastically with metal ions, this leads to steady current, which is directly 
proportional to applied voltage and this explains ohm‘s law. 
In the year 1909 Lorentz applied Maxwell – Boltzmann distribution law to 
electron gas to explain free electron theory of metals & hence it is known as 
Drude –Lorentz theory. 
The assumptions of free electron theory of metals: 
 
1. The free electrons in metal is treated as molecules of gas, hence it is assumed 
that they obey laws of kinetic theory of gases and in the absence of applied 
electric field energy associated with each electron. At temperature T is 3/2 kBT, 
which is equal to kinetic energy of electron 3/2 kBT = 1/2mv2, 
 vth= thermal velocity or r.m.s velocity 
2. The field due to positive ion is constant and force of repulsion between 
electrons is very small hence it is neglected. 
3. The flow of current in metal is due to applied field is consequence of drift 
velocity. 
Drift velocity 
The free electrons in metal are random in motion in the absence of electric field. 
However when electric field is applied, even though randomness persist but 
overall there is a shift in its position opposite to applied field. The electrons 
acquire kinetic energy and collide with ions, loses energy and this process 
continues until electrons acquire constant average velocity opposite to applied 
field. This velocity is called drift velocity. 
 
When electric field E is applied, electrons acquires constant average velocity Vd 
31 
 
If ‗m‘ is mass of electron and ‗e‘ is charge carried by electron then the resistive 
force Fr 
Acting on electron is equal to m a =
dmV
τ
 
 The driving force = - eE 
 
 Therefore 
dmV
τ
 = - e E τ = mean collision time 
 
 
τ


d
e
V
m
 
 
Summary: 
 
The free electrons in metal are responsible for most of the properties exhibited 
by metals 
In metal the absence of electric field electrons are random in motion 
The flow of current in metal is due to drift velocity 
 
LESSON-II 
 
Objectives 
 
At the end of this lesson we shall understand that: 
 The current flowing through it is directly proportional to drift velocity 
 The current density is directly proportional to electric field 
 
Introduction: 
 
In this lesson we will study some important definitions like relaxation time, 
mean free path and mean collision time and expression for current in a 
conductor.Relaxation time: 
“The time required for the average vector velocity to decrease to 1/e 
times its initial value when the field is just turned off is called the 
relaxation time”. 
Explanation: 
32 
 
 In a metal due to the randomness in the direction of motion of the conduction 
electrons, the probability of finding an electron moving in any given direction is 
equal to finding some other electron moving in exactly the opposite direction in 
the absence of an electric field. As a result the average velocity of the electrons 
in any given direction becomes zero. 
 i.e. vav = v'av , in the absence of the field 
However, when the metal is subjected to an external electric field, there will be 
a net positive value v'av for the average velocity of the conduction of the 
electrons in the direction of the applied field due to the drift velocity. 
 i.e. vav = v'av , in the presence of the field 
If the field is turned off suddenly, the average velocity vav reduces 
exponentially to zero from the value v'av which is also the value of vav just when 
the field is just turned off. 
 The decay process is represented by the equation (figure 2) 
 vav = v'av e-t /τr ----------------(1) 
Where t: time counted from the instant the field is turned off. 
 τr :constant called relaxation time 
 
 
Figure: Decay of average velocity 
 
 In equation (1) if t = τr , then 
 
 
1'  evv avav
 
33 
 
avav v
e
v '
1

 
Therefore the relaxation time can be defined as: 
“Consequent to the sudden disappearance of an electric field, across a 
metal, the average velocity of its conduction electrons decays 
exponentially to zero & the time required in this process for the average 
velocity to reduce to 1/e times its value just when the field is turned off is 
known as relaxation time”. 
Mean free path (λ) 
According to kinetic theory of gases, the mean free path is the distance traveled 
by a gas molecule in between two successive collisions. In the classical theory of 
free electron model, it is taken as the average distance traveled by the 
conduction electrons between successive collisions, with the lattice ions. 
Mean collision time (τ) 
“The average time that elapses between two consecutive collisions of an 
electron with the lattice points is called mean collision time”. The 
averaging is taken over a large number of collisions. 
If ‗v‘ is the velocity of the electrons which is the velocity due to the combined 
effect of thermal & drift velocities, then the mean collision time is given by, 
 
v

 
 
where λ is mean free path. 
 If vd is the drift velocity, then vd = vth 
 → v ≈ vth 
 In the case of metals, it can be shown that the relaxation time ‗t‘ always refer to 
a single total velocity called Fermi velocity. 
 
Mean free path 
 
 It is the average distance travelled by conduction electron between successive 
collisions. 
 
Mean free path is about 10-7 m. 
 
34 
 
Electric Field (E) 
The electric field is defined as potential per unit length of homogeneous 
conductor of uniform cross-section. i.e 
L
V
E 
volts/m 
If ‗L‘ is the length of a conductor of uniform cross – section & of uniform 
material composition & ‗V‘ is the potential difference between its two ends, then 
the electric field ‗E‘ at any point it is given by 
 
L
V
E 
 ---------------------- (2) 
Conductivity (σ): 
 It is the physical property that characterizes the conducting ability of a 
material. 
 If ‗R‘ is the uniform resistance of uniform material of length ‗L‘ & area of 
cross – section ‗A‘ then the electrical conductivity is given by 
 
A
L
R

1

--------------------(3) 
 If we consider the product σ E, then from (2) 
 
L
V
A
L
R
E 
1

 RA
V
E 
 
 Since I = V/R, we have A
I
E 
 --------------------- (4) 
 Comparing (1) & (4) we get 
 
EJ 
 ----------------------- (5) 
This represents the Ohm‘s law. 
Resistivity (ρ): 
 Resistivity signifies the resistance property of the material & is given by the 
inverse of conductivity. 
i.e 


1

---------------------- (6) 
35 
 
Expression for current in a conductor: 
 
 
Figure: Current carrying conductor 
 Consider a conductor of uniform area of cross-section ‗A‘, carrying a 
current ‗I‘ as shown in figure (5). If ‗v‘ is the velocity of the electrons, then the 
length traversed by the electron in unit time is ‗v‘. 
 Consider in the conductor, an imaginary plane at ‗X‘ normal to the 
current‘s direction. If we consider the motion of electrons as a group starting 
from ‗X‘, then they sweep a volume ‗vA‘ of the conductor in unit time as 
indicated in the figure. 
 Let ‗n‘ be the number of electrons/unit volume. Therefore 
 the number of electrons in a volume 
)(vAnvA 
 
In other words, the number of electrons crossing any cross-section in unit time 
= 
)(vAn
 
If ‗e‘ is the charge on each electron crossing any section per second is the same 
as rate of flow of charge. Therefore 
 the rate of flow of charge = 
nevA
--------------------(1) 
Since the velocity acquired by the electron is due to an applied electric field, it is same 
as drift velocity 
dv
 i.e 
dvv 
 

 
AvenI d
 -------------------- (2) 
Expression for electrical conductivity: 
Consider the motion of an electron, in a conductor subjected to the influence of 
an electric field. If ‗e‘ is the charge on the electron and ‗E‘ is the strength of the 
applied field, then the force ‗F‘ on the electron is EeF 
 ---------------------- (1) 
36 
 
If ‗m‘ is the mass of an electron then as per Newton‘s second law of motion force 
on the electron can be written as dt
dv
mF 
 ----------------------(2) 
Equating (1) & (2) dt
dv
mEe 
 

 
dt
m
eE
dv
 
 Integrating both sides, 
 
dt
m
eE
dv
t
 
0
 
t
m
eE
v
---------------------(3) 
where t: time of traverse 
Let the time of traverse ‗t‘ be taken equal to the collision time ‗τ‘. Since by 
definition the collision time applies to an average value, the corresponding 
velocity in (3) also becomes the average velocity v. 
 
m
eE
v 
 
We have the expression for electrical conductivity ‗σ‘ as E
J

 --------------------- (5) 
Where ‗J‘ is the current density & J = I /A, where ‗I‘ is the in the conductor & 
‗A‘ is the area of cross-section of the conductor. 
  AE
I

 --------------------- (6) 
Now the distance covered by electrons in a unit time is v. They sweep a volume 
equal to vA (figure) in a unit time. If ‗e‘ is the charge on the electron, ‗n‘ is the 
37 
 
number of electrons per unit volume, then the quantity of charge crossing a 
given point inthe conductor per unit area per unit time i.e the current ‗I‘ is 
 I = 
nevA
-------------------(7) 
Substituting for ‗I‘ from (7) in (6) we get 
 
EA
Aven

 
 
E
ven

 -------------------- (8) 
Substituting for ‗v‘ from (4) in (8) we get 
 
 






m
Ee
E
en
 
 
m
en 

2

 ----------------------(9) 
Equation (9) is the expression for electrical conductivity of a conductor. 
Mobility of Electrons 
The mobility of electrons is defined as the magnitude of the drift velocity 
acquired by the electrons in unit field. Thus if ‗E‘ is the applied electric field, in 
which the electrons acquire a drift velocity vd , then the mobility of electrons 
 
E
vd
 --------------------(1) 
We have for Ohm‘s law, 
 
EJ 
 
 Hence, E
J

 
 en
EA
ven d 
 
38 
 
 
en

 
 ---------------------- (2) 
 We know that, m
en 

2

 
Therefore (2) becomes, 
 enm
en 1
2



 
Hence, 
m
e
 
 --------------------- (3) 
 Mobility represents the ease with which the electrons could drift in the 
material, under the influence of an electric field. Different materials have 
different values for mobility. 
Failure of classical free electron theory 
Though the classical free electron theory has been successful in accounting for 
certain important experimental facts such as electrical and thermal 
conductivities in metals: it failed to account for many other experimental facts 
among which the notable ones are specific heat, temperature dependence of σ 
and the temperature dependence of electrical conductivity on electron 
concentration. 
(1) Specific Heat: 
 The molar specific heat of a gas at constant volume is 
 
R
2
3
Cv 
 
 As per the classical free electron theory, free electrons in a metal are 
expected to behave just as gas molecules. Thus the above equation holds good 
equally well for the free electrons also. But experimentally it was found that, the 
contribution to the specific heat of a metal by its conducting electrons was 
smaller than the classical value (3/2)R by a factor of about 10-2 . 
(2) Temperature dependence of electrical conductivity: 
39 
 
 It has been observed that for metals the electrical conductivity ‗σ‘ is 
inversely proportional to the temperature ‗T‘. 
 i.e 
T
1
exp 
-----------------(1) 
But according to the main assumptions of classical free electron theory 
 
th
2vm
2
1
TK
2
3

 
 
m
KT3
v th
2 
 
 
 
Tv th 
 
 
Since the mean collision time τ is inversely proportional to the thermal velocity, 
we can write, 
thv
1

 
 
Or T
1

 ------------------- (2) 
But σ is given by, 
 
m
en 

2

 
Therefore the proportionality constant between σ and τ can be represented as 
 
 
 
or T
1

-------------------- (3) [From proportionality (2)] 
Now from the proportionality representations (1) & (3), it is clear that the 
prediction of classical free electron theory is not agreeing with the experimental 
observations. 
3. Dependence of electrical conductivity on electron concentration: 
40 
 
 
 
 As per the classical free electron theory, the electrical conductivity ‗σ‘ is 
given by, 
 
m
en 

2

 
where n: concentration of the electrons, therefore 
 σ α n 
If we consider the specific cases of Zinc and Cadmium which are divalent 
metals, the electrical conductivities are respectively 1.09X 107 / Ω m & 0.15 X 
107 / Ω m which are much lesser than that for Copper and Silver, the values for 
which are 5.88 X 107 / Ω m & 6.3 X 107 / Ω m respectively. On the other hand, 
the electron concentration for Zinc and Cadmium are13.10 X 1028 /m3 & 9.28 
X 1028 /m3 which are much higher than that for Copper and Silver, the 
values for which are 8.45 X 1028 /m3 and 5.85 X 1028 /m3 respectively. 
 
LESSON-3 
Objectives: 
 At the end of this lesson we shall understand that: 
 All the free electrons cannot receive energy 
 The electrons are not completely free as assumed in classical theory 
 The free electrons obey Pauli‘s exclusion principle 
 The energy levels of an electron in a metal are quantized 
Introduction: 
In this lesson, we will study Quantum free electron theory of metals, Fermi –
Dirac distribution function and Fermi energy and Fermi factor, 
41 
 
Quantum Free Electron Theory of Metals 
One of the main difficulties of classical free electron theory of metals is that, it 
allows all the free electrons to gain energy (as per Maxwell-Boltzmann 
statistics). Hence value obtained are much higher than experimental values 
After development of quantum statistics, it is realized that only one percent of 
free electrons can thus absorb energy . This brings importance of Pauli‘s 
exclusion principle. 
In the year 1928 Arnold Summerfeld applied quantum mechanical conditions 
and Pauli‘s exclusion principle to explain failures of classical free electron 
theory of metals. 
 
The following are some of the assumptions of quantum free electron theory: 
1) The energy levels of free electrons in metals are quantized 
2) The free electrons obey Pauli Exclusion Principle 
3) The electrons travel in side metal with constant velocity, but they are 
confined within boundaries 
4) The distribution of electrons are among various levels as per Fermi-Dirac 
statistics. 
5) The force of attraction between electrons and positive ionic lattice also force 
of repulsion among electrons is neglected 
Fermi–Dirac Statistics 
According to Sommerfeld the electrons are not completely free in the metal as 
predicted in the free electron theory, i.e they are partially free and bound to the 
metal as a whole, hence electrons in metal can ot be compared to gas 
molecules; therefore, we cannot apply Maxwell-Boltzmann statistics. Moreover 
electrons are assumed to obey Pauli‘s exclusion principle hence they are 
governed by Fermi –Dirac statistics . 
The electrons obeying Pauli‘s exclusion principle are identical and 
indistinguishable particles called ‗Fermions. The Fermi–Dirac distribution 
function gives most probable distribution of electrons. Hence in equilibrium at 
a temperature T, the probability that an electron has an energy E is given Fermi 
function f (E). 
E is energy level whose occupancy is being considered, EF = Fermi level, it is 
constant for a particular metal. 
At absolute zero f(E) = 0 for E >EF and f (E) =1 for E< E F 
The Fermi level is highest state for the electrons to occupy at absolute zero, that 
mean at absolute zero Fermi level divides the occupied states from the 
unoccupied states. 
42 
 
Fermi Energy 
In quantum free electron theory, the energy of electron in metal is quantized, 
therefore according to quantization rules, if there are N numbers of electrons, 
then there must be N number of allowed energy levels, since these electrons 
obey Pauli‘s exclusion principle. 
An energy level can accommodate at most only two electrons, one with spin up 
and other with spin down, thuswhile filling energy levels, two electrons occupy 
the least level, two more next level and so forth, until all electrons are 
accommodated as shown in figure. 
 The energy of the highest occupied level at absolute zero temperature is called 
Fermi Energy and the corresponding energy level is called Fermi Level. The 
Fermi energy can also be defined as maximum kinetic energy possessed by free 
electron at absolute zero temperature, it is denoted by EF. 
 
At absolute zero temperature, metal does not receive energy from surroundings, 
therefore all the energy levels below Fermi level is completely filled up and 
above the Fermi level all the energy levels are empty, If there are N electrons in 
the metal then highest occupied level is N/2 this level is called Fermi level and 
corresponding energy is called Fermi energy. 
Fermi Factor 
When the temperature is greater than the 0 K, metal receives thermal energy 
from the surroundings; however, at room temperature thermal energy received 
by metal is very small (kBT = 0.025 eV, kB = Boltzmann‘s constant), hence the 
electrons in the energy levels far below Fermi level cannot absorb this energy 
because there are no vacant energy levels above them, however the electrons 
just below Fermi level absorb this energy and may move to unoccupied energy 
levels above Fermi level, though these excitations seems to be random, the 
occupation of various energy levels takes place strictly as per Fermi –Dirac 
distribution law. 
Fermi function, f (E) = 1/(1+e
(E-E
F
)/kT
), where f (E) is the probability of an electron 
occupying energy state E. 
 
43 
 
 
(i) For T = 0K and E > EF 
 
f (E) = 1/(1+e
∞) = 1/∞ = 0. ie no electron can have energy greater than EF at 
0K. 
 
(ii) For T = 0K and E < EF 
 
f (E) = 1/(1+e
-∞
) = 1/1+0 = 1. ie all electrons occupy energy states below EF 
at 0K. 
 
(iii)For T > 0K and E = EF 
 
f (E) = 1/(1+e
0
) = 1/1+1 = ½. ie 50% electrons can occupy energy states 
below EF 
 
above 0K. 
 
Fermi level is defined as energy level at which the probability of electron 
occupation is one half or 50 %. 
Summary of Lesson 
Here we learnt that, in metal, free electrons are partially free, because they are 
bound to the metal as a whole, hence they cannot be compared to molecules of 
gas. 
In metal there is extremely larger number of energy levels. The distribution of 
electrons among various energy levels is strictly as per Fermi -Dirac function. At 
0 K all the energy levels below Fermi level is occupied and above Fermi level, 
energy levels are empty. 
LESSON 4 
Objectives 
At the end of this lesson we shall understand that: 
 The free electrons in metal can be treated as particles in a box 
 The Fermi temperature is only theoretical concept 
 The total energy of free electron is 3/5 EF 
44 
 
Introduction 
In this lesson, we will study number of available energy states in the range E 
and E + dE, Number of electrons per unit volume, Fermi energy, Fermi 
temperature and Fermi velocity. 
Density of states 
The electron energy levels in a material are in terms of bands. The number of 
levels in each band is extremely large and these energy levels are not evenly 
distributed in the band. At the highest energy the difference between 
neighboring levels is of the order of 10-6 eV. That means in a small energy 
interval dE there are still many discrete energy levels. Hence for easy 
calculations we introduce the concept of ‗Density of States‗. It is denoted by g 
(E). The density of state can be defined as follows: 
‗It is the number of available states per unit volume per unit energy range‘. 
Number of available states per unit volume between energy range, 
E and E+dE= g (E) dE. 
Summary 
Here we learnt that, in metal, free electrons are partially free, because they are 
bound to the metal as a whole, hence they cannot be compared to molecules of 
gas. In metal there is extremely larger number of energy levels. The distribution 
of electrons among various energy levels is strictly as per Fermi -Dirac function. 
At 0 K, all the energy levels below Fermi level are occupied and above Fermi 
level, energy levels are empty. 
LESSON-5 
Objectives 
At the end of this lesson we shall understand that: 
The electron moving in metal under influence of external field possesses 
effective mass. The effective mass varies from solid to solid and it is a function 
of energy. 
The drift velocity of free electrons in metal is equal to Fermi velocity. 
Introduction 
In this lesson we study merits of Quantum free electron theory. 
Concept of effective mass 
According to Sommerfeld, Quantum free electron theory of metals, the motion 
of free electrons in metals is considered not as motion of particles, but as 
passage of waves among periodic lattice. Hence the motion of electrons in metal 
can be treated as a wave packet. Hence velocity of electron is treated as group 
45 
 
velocity. When field is ,applied to an electron; the wave packet travels under 
combined action of applied field and potential due to periodic lattice and due to 
this superposition of these fields the electron responds as if it posses effective 
mass ; this mass is different from its true mass with which it moves under the 
influence of external field alone. The effective mass is interpreted in terms of 
true mass. The Concept of Effective Mass shows that it is possible to deal with 
the motion of electrons in metal as semi classical manner. In vacuum the 
effective mass of electron is same as true mass and the Effective Mass varies 
from solid to solid. 
Electrical Resistivity or Conductivity 
According to Sommerfeld, Quantum free electron theory of metals the electrons 
are partially free not completely as assumed in classical free electron theory. 
Hence free electrons in metals obey Fermi –Dirac statistics. By applying 
Boltzmann transport equation and Fermi –Dirac statistics he got the equation 
for electrical conductivity of metals as, 
 In classical free electron theory it is assumed that Electrical Resistivity in 
metals is due to scattering of electrons and the scattering of electrons takes 
place due to lattice defects, dislocations, impurities etc, but according to 
Summerfield the motion of electrons in metals nothing but passage of waves 
in periodic lattice, if there is perfect periodicity and all ions are at rest, then the 
waves pass across the arrays without being scattering at all in such case 
mean free path is infinite, but no metal is free from impurities or lattice 
defects, that mean there is always deviations from periodicity due to this, 
scattering of electron waves takes place, therefore lattice defects becomes 
major cause of electrical resistivity in metals and scattering of electron waves 
becomes the deciding factor for the mean free path of electrons. 
 
Merits of quantum free electron theory 
(1) Specific Heat : 
According to quantum free electron theory, it is only those electrons that are 
occupying energy levels close to EF which are capable of absorbing the heat 
energy to get excited to higher energy levels. Thus only a small percentage of 
the conduction electrons are capable of receiving the thermal energy input and 
hence the specific heat value becomes very small for the metals. 
Therefore on the basis of quantum free electron theory 
 
TR
E
k
C
F
B
V 






2
 
Taking a typical value of EF = 5eV we get 
 
410
2 





F
B
E
k
 
46 
 
  CV = 10-4 RT 
This agrees with the experimental values. Since CV is very small, the energy of 
electrons isvirtually independent of temperature. 
 
(2) Temperature dependence of electrical conductivity: 
 The experimentally observed fact that electrical conductivity ‗σ‘ has a 
dependence on 






T
1
but not on 
T
1
 can be explained as follows. 
 The expression for electrical conductivity is given by 
 


m
en F
2
 
 As per quantum free electron theory, 
F
F
F
v


 
 
F
F
vm
en




2
 ---------------- (1) 
As per quantum free electron theory EF and vF are essentially independent of 
temperature. But λ F is dependent on temperature, which is explained as 
follows. 
As the conduction electrons traverse in the metal, they are subjected to 
scattering by the vibrating ions of the lattice. The vibrations occur such that the 
displacement of ions takes place equally in all directions. If ‗a‘ is the amplitude 
of vibrations, then the ions can be considered to present effectively a circular 
cross- section of area Πa2 that blocks the path of the electrons irrespective of 
the direction of approach. Since vibrations of larger area of cross-section should 
scatter more efficiently, it results in a reduction in the value of mean free path 
of the electrons, 
 
2
1
a
F 
 
 ------------------ (2) 
Considering the facts that, 
(a) the energy of a vibrating body is proportional to the square of the 
amplitude 
(b) the energy of ions is due to thermal energy 
(c) thermal energy is proportional to the temperature (T). 
47 
 
 Therefore we can write, 
 
Ta 2
 
 
TF
1

 ------------------ (3) 
From (1) & (3) we get, 
 







T
1

 
Thus the dependence of ‗σ‘ on ‗T‘ is correctly explained by the quantum free 
electron theory. 
(3) Electrical conductivity and electron concentration: 
By classical free electron theory, it was not possible to understand why metals 
such as Al and Ga which have 3 free electrons per atom have lower electrical 
conductivity than metals such as copper and silver which possess only one free 
electron per atom. But according to quantum free electron theory the same can 
be explained. We have the equation for electrical conductivity as: 
 
F
F
vm
en




2
 
From this equation it is clear that, the value of σ depends on both ‗n‘ and the 
ratio F
F
v

 . If we compare the cases of copper and aluminum, the value of ‗n‘ 
for Al is 2.13 times higher than that of copper. But the value of F
F
v

 for copper 
is about 3.73 times higher than that of Al. Thus, the conductivity of copper 
exceeds that of aluminum. 
Comparison between Classical free electron theory and quantum free electron 
theory 
 Similarities: 
1) The valence electrons are treated as though they constitute an ideal gas. 
2) The valence electrons can move easily throughout the body of the solid. 
48 
 
3) The mutual repulsion between the electrons and the force of attraction 
between electrons and ions are considered insignificant. 
 
Differences between the two theories: 
Sl.no Classical free electron 
theory 
Quantum free electron theory 
 1. The free electrons which 
constitute the electron gas 
can have continuous 
energy values. 
The energy values of the free electrons are 
discontinuous because of which their 
energy levels are discrete. 
 2. It is possible that many 
electrons may possess 
same energy. 
The free electrons obey the ‗Pauli‘s 
exclusion principle‘. Hence no two 
electrons can possess same energy. 
 3. The patterns of 
distribution of energy 
among the free electrons 
obey Maxwell-Boltzmann 
statistics. 
The distribution of energy among the free 
electrons is according to Fermi-Dirac 
statistics, which imposes a severe 
restriction on the possible ways in which 
the electrons absorbs energy from an 
external source. 
 
Solved Problems 
1. Calculate the drift velocity and thermal velocity of free electrons in copper at room 
temperature, (300 k), when a copper wire of lengths 3 m and resistance 0.022 carries of 
15 A. 
 Given:  = 4.3 × 10-3 m2 /Vs. 
Solution: Given that L= 3 m, R =0.022, I= 15 A, T = 300 k, cu = 4.3 × 10
-3 m2 /Vs. Vd =? and V th 
= ? 
 Voltage drop across the copper wire is given by 
 V= IR = 15 × 0.022 = 0.33 V 
 Electric Field, E = V / L = 0.33 /3 = 0.11 V /m 
 Drift velocity, Vd = E ×  = 0.11 ×4.3 × 10
-3 = 0.473 × 10-3 m/s. 
 Thermal Velocity, Vth = 3kT / m 
 = 3 × 1.387 × 10-23 ×300 / 9.11×10-31 
 = 1.17 ×105 m/s. 
49 
 
2. Find the relaxation time of conduction electrons in a metal of resistivity 
1.54 x 10-8 ohm-m, if the metal has 5.8 x 1028 electrons /m3 
Given ρ = 1.54 x 10-8 ohm-m n = 5.8 x 10-8 electrons /m3 
Resistivity of metal = ρ = = 
Relaxation time = ρ = 3.97 x 10-14 s 
 
 
CONDUCTIVITY IN SEMICONDUCTORS 
LESSON-6 
Objectives: 
 To study the semiconductor energy level diagram 
 To derive an expression for hole and electron concentration. 
 To observe the Hall Effect in a semiconductor. 
Based on the electrical conductivity of the materials they can be classified into 
three categories. Conductors- conductors are the materials that allow the 
electricity to pass through them. Eg: aluminum, copper, silver, etc. Insulators- 
insulators are the materials that do not allow the electricity to pass through 
them. Eg: paper, glass, etc. 
Semiconductors- semiconductors are materials whose electrical conductivity 
lies between that of conductors and insulators. Eg: silicon and germanium. 
Conductivity in semiconductors: 
Atoms of silicon and germanium have four electrons in their outer most shell. 
These electrons form covalent bond with the neighbouring atom and not free at 
low temperature. Hence they behave like insulators. However when a small 
amount of thermal energy is available from the surroundings a few covalent 
bonds are broken and few electrons are set free to move. Even at room 
temperature good number of electrons is dissociated from their atoms and this 
number increases with rise in temperature. This leads to conductivity. When an 
electron is detached from the covalent bond, it leaves a vacancy which behaves 
like a positive charge. An electron from a neighbouring atom can move onto this 
vacancy leaving a neighbor with a vacancy. Such a vacancy is called a hole. 
Hole acts as a positive charge. 
Types of semiconductors: 
50 
 
In a semiconductor there are two kinds of current carriers- Electrons and Hole. 
In a pure semiconductor electrons and holes are always present in equal 
numbers and it is called intrinsic semiconductor. 
Conductivity of the semiconductors can be changed by adding small amount of 
impurities (other elements) to it. These impurities are called dopants. Such 
semiconductors are impure or extrinsic semiconductors. When a few atoms of 
trivalent or pentavalent element is added into pure crystals of Ge or Si an 
extrinsic semiconductors are produced. The process of adding impurity atoms 
is called Doping. When pentavalent impurity atoms like arsenic, antimony, 
phosphorous, etc are added to pure germanium or silicon crystal, we get an 
extrinsic semiconductor known as n-type semiconductor. When trivalent 
impurity atoms like indium, boron, gallium, aluminum, etc are added to pure 
germanium or silicon crystal, we get an extrinsic semiconductor known as p-
type semiconductor. 
 
Concentration of electrons