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UNIT-III

QUANTUM MECHANICS AND ELECTRON THEORY

Basic terms and definitions

Matter waves Wave associated with the motion of a particle (or) Matter waves are nothing but waves and are usually associated with a moving quantum particle.

de Broglie hypothesis Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter can exhibit wave-like behaviour. The concept that matter behaves like a wave was proposed by Louis de Broglie in 1924. It is also referred to as the de Broglie hypothesis.

Mobility In steady state the drift velocity per unit electric field is called Mobility.

Relaxation Time The time taken for the drift velocity of an electron to decay 1/e times of its initial value is called Relaxation Time

Mean collision time The duration between two successive collisions when electron is in random motion is called Mean collision time

Mean free path The average distance travelled by an electron between two successive collisions during their random motions is called mean free path

Drift Velocity

When an electric field (E) is applied to the metal electrons experience force (F) in opposite direction to the applied field. this force is called Drift Velocity

Merits of Classical free electron 1.It verifies ohm's law 2.It explains the electrical and thermal conductivities of metals. 3.It derives Weid eman-franz law. 4.It explains optical properties of metal.

Postulates of Classical free electron theory

1. In metals there are large number of free electrons moving freely in all possible directions 2. free electrons in the metals are assumed to behave like gas molecules obeying the kinetic theory of gases 3. In metals electrons move randomly and collide with either +ve ions or with free electrons all the collisions are elastic (i.e., no lose of energy) 4. When electron field is applied to the metal the free electrons accelerated in the opposite direction to the applied field

Drawbacks of Classical free electron

1.Classical free electron fails to explain the temperature dependence of electrical conductivity 2 .fails to explain specific heat of a solid. 3. fails to explain Wiedemann-franz law that is at lower temperatures, the ratio of thermal conductivity to electrical conductivity is proportional to temperature. 4. Electrical conductivity for semiconductors and insulators are not explained.

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Postulates of quantum free electron theory

1 .In a metal the available free electrons are fully responsible for electrical conduction. 2. The electrons move in a constant potential inside the metal.. 3. Electrons have wave nature, the velocity and energy distribution of the electron is given by Fermi-Dirac distribution function. 4. The loss of energy due to interaction of the free electron with the other free electron. 5.Electron’s distributed in various energy levels according to Pauli Exclusion Principle.

Advantages of Quantum free electron theory

1.This theory explains the specific heat capacity of materials. 2.This theory explains photo electric effect, Compton Effect and block body radiation. etc. 3.This theory gives the correct mathematical expression for the thermal conductivity of metals.

Drawbacks of Quantum free electron theory

1.This theory fails to distinguish between metal, semiconductor and Insulator. 2.It also fails to explain the positive value of Hall Co-efficient. 3.According to this theory, only two electrons are present in the Fermi level and they are responsible for conduction which is not true.

Fermi Level The Fermi Level is the highest energy level which an electron can occupy at the absolute zero temperature.

Fermi energy The Fermi energy is the energy of the highest level of quantum state which is occupied by the fermions (like electrons, protons or neutrons) at the absolute zero temperature.

Sources of electrical resistances 1.Lattice vibration 2.Impurities 3.Structural imperfection

Energy band Energy bands consisting of a large number of closely spaced energy levels exist

Energy Gap The gap between conduction band and valence band. It is denoted by Eg. Energy gap value is different for different solids. Energy gap in Metals-0 ev

Energy gap in semiconductors ev1

Energy gap in Insulators - ev5

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Concepts:

Quantum Mechanics:

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Matter wave Properties:

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Physical significance of :

The wave function 𝛙 enables all possible information about the particle. 𝛙 is a complex quantity and has no direct physical meaning. It is only a mathematical tool in order to represent the variable physical quantities in quantum mechanics.

2. Born suggested that, the value of wave function associated with a moving particle at the position co-ordinates (x,y,z) in space, and at the time instant ‘t’ is related in finding the particle at certain location and certain period of time ‘t’.

3. If 𝛙 represents the probability of finding the particle, then it can have two cases. Case 1: certainty of its Presence: +ve probability Case 2: certainty of its absence: - ve probability, but –ve probability is meaningless, Hence the wave function 𝛙 is complex number and is of the form a+ib

4. Even though 𝛙 has no physical meaning, the square of its absolute magnitude |𝛙2| gives a definite meaning and is obtained by multiplying the complex number with its complex conjugate then |𝛙2| represents the probability density ‘p’ of locating the particle at a place at a given instant of time. And has real and positive solutions.

𝛙 (𝐱,𝐲,𝐳,𝐭)=𝐚+𝐢𝐛 𝛙∗(𝐱,𝐲,𝐳,𝐭)=𝐚−𝐢𝐛 𝐩= 𝛙𝛙∗=|𝛙2|=𝑎2+𝑏2 𝑎𝑠 𝑖2=−1 Where ‘P’ is called the probability density of the wave function

5.If the particle is moving in a volume ‘V’, then the probability of finding the particle in a volume element dv, surrounding the point x,y,z and at instant ‘t’ is Pdv

∫|𝛙2|𝑑𝑣=1 𝑖𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑒𝑙 𝑖𝑠 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 ∫|𝛙2|𝑑𝑣 = 0 if particle does not exist

This is called normalization condition

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Particle in one dimensional Potential Box:

The wave nature of a moving particle leads to some remarkable consequences when the

particle is restricted to a certain region of space instead of being able to move freely .i.e when

a particle bounces back and forth between the walls of a box.

The wave nature of a moving particle leads to some remarkable consequences when the

particle is restricted to a certain region of space instead of being able to move freely .i.e when

a particle bounces back and forth between the walls of a box.

The Schrodinger wave equation will be applied to study the motion of a particle in 1-D box to

show how quantum numbers, discrete values of energy and zero point energy arise.

From a wave point of view, a particle trapped in a box is like a standing wave in a string

stretched between the box’s walls.

Consider a particle of mass ‘m’ moving freely along x- axis and is confined between x=0 and x=

a by infinitely two hard walls, so that the particle has no chance of penetrating them and

bouncing back and forth between the walls of a 1-D box.

If the particle does not lose energy when it collides with such walls, then the total energy

remains constant.

This box can be represented by a potential well of width ‘a’, where V is uniform inside the box

throughout the length ‘a’ i.e V= 0 inside the box or convenience and with potential walls of

infinite height at x=0 and x=a, so that the PE ‘V’ of a particle is infinitely high V=∞ on both

sides of the box.

The boundary condition are

𝑣(𝑥)=0 ,𝜓(𝑥)=1𝑤ℎ𝑒𝑛 0<𝑥<𝑎…. (1)

𝑣(𝑥)=∞ ,𝜓(𝑥)=0𝑤ℎ𝑒𝑛 0≥𝑥≥𝑎… (2)

Where 𝜓(𝑥) is the probability of finding the particle.

The Schrodinger wave equation for the particle in the potential well can be written as

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Electron Theory of Metals:

The electron theory of solids explains the structures and properties of solids through their

electronic structure. This theory is applicable to all solids both metals and non metals. This theory

also explains the bending in solids behavior of conductors and insulators, electrical and thermal

conductivities of solids, elasticity and repulsive forces in solids etc,. The theory has been developed

in three main stages.

Classical free electron theory

This theory was developed by Drude and Lorentz. According to this theory, a metal consists of

electrons which are from to move about in the crystal molecules of a gas it contains mutual

repulsion between electrons is ignored and hence potential energy is taken as zero Therefore the

total energy of the electron is equal to its kinetic energy.

A Solid metal has nucleus with revolving electrons. The electrons have freely like

molecules in a gas

In the absence of electric field (E=0), the free electrons move in random directions

and collide with each other. During this collision no loss of energy is observes since

the collisions are elastic

When the presence of electric field the free electrons are accelerated on the

direction opposite to the applied electric field

Since the electrons are assumed to be perfect gas, they obey the laws of classical theory of gases

Classical free electrons in the metal obey Maxwell-Boltzmann statistics.

EXPRESSION FOR ELECTRICAL CONDUCTIVITY The electrical conductivity is defined as the quantity of electricity flowing per unit area per unit time at a constant potential gradient.

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When an electric field (E) is applied to a conductor the free electrons are accelerated and give

rise to current (I) which flows in the direction of electric filed flows of charges is given terms of

current density.

Let ‘n’ be the numb r of electrons per unit volume and ‘e’ be the charge of the electrons

The current flowing through a conductor per unit area in unit time (current density) is given by

J = nVd ( e)

J = – nV d (e) ... (1) The negative sign indicates that the direction of current is in opposite direction to the movement of electron.

Due to the applied electric field, the electrons acquire an acceleration ‘a’ can be given by

a= Vd/ τ

Vd=a τ ……. (2)

When an electric field of strength (E) is applied to the conductor, the force experienced by the free

electrons in given by F=-eE …..(3)

From Newton’s second Law of motion, the force acquired by the electrons can be written as

F = ma ….(4)

Comparing equation (3) & (4)

–eE = ma

a = -eE/m

Now, substituting the value of ‘a’ from the eq tion (2),we get

----------(6)

Substitute equation (6) in (1)

m

EneJ

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The electrical conductivity =

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QUANTUM FREE ELECTRON THEORY

The failure of classical free electron theory paved this way for Quantum free electron theory. It as

introduced by Sommerfield in 1928. This theory is based on making small concepts. This theory was proposed

by making small changes in the classical free electron theory and by retaining most of the postulates of the

classical free electron theory.

Assumptions (Postulates) of Quantum free electron theory

1. In a metal the available free electrons are fully responsible for electrical conduction.

2. The electrons move in a constant potential inside the metal. They cannot come out from the metal surface

have very high potential barrier.

3. Electrons have wave nature, the velocity and energy distribution for the electrons given by Fermi-Dirac

distribution function.

4. The loss of energy due to interaction of the free electron with the other free electron

5. Electron’s distributed various energy levels according to Pauli Exclusion Principle

Advantages of Quantum free electron theory

1. This theory fails to distinguish between metal, semiconductor and Insulator

2. It also fails to explain the positive value of Hall Co-efficient.

3. According to this theory, only two electrons are present in the Fermi level and they are responsible for

conduction which is not true.

FERMI – DIRAC DISTRIBUTION FUNCTION:

It is an expression for the distribution of electrons among the energy levels as a function of temperature, the probability of finding an electron in a particular energy state of energy E is given by

Where EF - Fermi energy (highest energy level of an electron)

KB - Boltzmann’s constant T - Absolute temperature AT=0K and E< EF

F(E)=1

1+exp(−∞)=

1

1+0

F(E)=1=100%

It means that 100% probability for the electrons to occupy the energy level below the Fermi energy level.

AT=0K and E> EF

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F(E)=1

1+exp(∞)=

1

0=0

F(E)=0=0%

It means that 0% probability (electron) for the electrons to occupy the energy level above the Fermi energy

level. AT=0K and E=EF

F(E)=1

1+exp(0)=

1

1+1=

1

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F(E)=0.5=50%

It means that 50% probability for the electrons to occupy the Ferm i energy level. (above Fermi energy level are empty and below Fermi energy level are filled). At 0 K energy states above EF are empty and below EF are filled.

Fermi level, Fermi Energy and their importance

Fermi energy level is defined as the highest reference level of a particle at absolute 0K

Importance: It is the reference energy level which separates the filled energy levels and vacant energy levels.

Fermi energy (EF) : The Fermi energy is the maximum energy of the quantum state corresponding to Fermi energy level at absolute zero

Importance: Fermi energy determines the energy of the particle at any temperature

Source of Electrical Resistance:

Lattice imperfections are the scattering centers for the origin of electrical resistance. The lattice

imperfections are

1. Thermal Vibrations at high temperatures

2. Impurities and defects at low temperatures

TiT

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KRONING-PENNY MODEL (Motion of an electron in periodic potential):

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Brillouin Zones:

Brillouin Zones are the permissible values of k of an electron moving in one, two, three dimensions in a

periodic potential. Thus the energy spectrum of an electron moving in a periodic potential consists of allowed

bands which are separated by forbidden bands.

The energy bands of permissive values of k are called Brillouin Zones

If k= -a

to

a

first Brillouin Zone

If k= -a

2 to

a

2 Second Brillouin Zone

If k= -a

3 to

a

3 Third Brillouin Zone

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Origin of Energy Bands in Solids:

An isolated atom possesses discrete energies of different electrons. Suppose two isolated atoms are brought to

very close proximity, then the electrons in the orbits of two atoms interact with each other. So, that in the

combined system, the energies of electrons will not be in the same level but changes and the energies will be

slightly lower and larger than the original value. So, at the place of each energy level, a closely spaced two

energy levels exists. If ‘N’ number of atoms are brought together to form a solid and if these atoms’ electrons

interact and give ‘N’ number of closely spaced energy levels in the place of discrete energy levels, it is known as

bands of allowed energies.

Classification of solids into metals, semiconductors and insulators:

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Examples: Al, Cu, Silver, Gold

Examples: Silicon 1.1 ev, Germanium 0.7ev

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IMPORTANT QUESTIONS:

1. Explain de-Broglie hypothesis. Derive an expression for de-Broglie wave length.

2. Show that the wavelength λ associated with an electron of mass ‘m’ and kinetic energy ‘E’ is given by

λ=ℎ

√2𝑚𝐸.

3. Show that the wavelength of an electron accelerated by a potential difference’V’ volts is λ=(1.227 X 10-

10) /√v m for non-relativistic case.

4. What are matter waves? Explain their properties.

5. Derive time independent Schrodinger’s wave equation for a free particle.

6. Derive time dependent Schrodinger’s wave equation for a free particle

7. Explain the physical significance of wave function.

8. Derive an expression for the energy of a particle in one- dimensional potential box of width ‘a’.(OR)

Show that the energies of a particle in a potential box are quantized.

9. Explain the Fermi-Dirac distribution function of electrons. Explain the effect of temperature on the

distribution.

10. What are the salient features of classical free electron theory (or) free electron gas model? Obtain an

expression for electrical conductivity.

11. What are the salient features of Quantum free electron theory? Obtain an expression for electrical

conductivity.

12. Discuss the origin (or) source of electrical resistance in solids.

13. Explain the origin of energy bands in solids.

14. Discuss the Kronig Penny model for the motion of an electron in a periodic potential.

15. Distinguish between metals, semi conductors and insulators