**Free Electron Theory of Metals**

A module of Solid State Physics Syllabus approved by Bangalore University

Just 3 years after the discovery of electron in 1897 by Thomson, Drude and Lorentz, motivated by the success of Kinetic Theory of Gases, employed the same to understand the properties of electrons in Metals. It was quite an achievement at the time, noting that it was before the discovery of Nucleus (Alpha scattering experiment by Rutherford, 1909) or a realistic picture of Atom, they could explain the resistivity of some metals.

Although the model was with numerous shortcomings and a Happy accident (Weideman Franz Law), it is a crucial starting point of our understanding macroscopic properties (Conductivity, thermal and Electrical) from a model describing the material structure.

With development of Quantum Mechanics, Fermi Statistics was incorporated, and the model was improved further, however, still considering electrons as free particles. Till this point, its the story of free electrons (classical and quantum), and the subject matter of the notes provided here.

Later developments gradually relaxed this condition of electrons being free and took into account a small interaction of electrons with the periodic lattice of metal, thus giving rise to Nearly Free Electron Model. On the other extreme, scientists started looking at tight-binding models in which electrons were tightly bound to their respective atoms but could jump to their nearest neighbors. These models gave rise to band theory of electrons in solids, and explained the properties that free electron theories could not.

## Notes 0

Introductory (Historical) Lecture on development of the field of Condensed Matter Physics and its evolution with development of Quantum Mechanics.

## Notes 1

1) Derivation of Electrical Conductivity in the frame of classical theory of free electrons

2) Derivation of Thermal Conductivity from free electron theory of metals.

## Notes 2

## Notes 3

1) Density of States (Different Methods)

2) Fermi Dirac statistics

3) Free Electron density at absolute zero

## Notes 4

1) Recap of DoS, Fermi Statistics

2) Average Energy of Free Electrons at T = 0 Kelvin

3) Basics of Hall Effect

## Polarization of Light

A module on Electromagnetism approved by Bangalore University

Polarization of light (Image Source)

Syllabus Covered:

- Introduction and Basic Definitions
- Methods of Polarization: By Reflection, Absorption, Dichroism, Birefringence
- Elliptically And Circularly Polarized Light and their production
- Laurent’s Half-Shade Polarimeter (Experimental Demonstration)

Lecture Notes on Polarisation

**Methods of Polarisation – I:**

Polarisation by Reflection, Refraction, Selective Absorption, Birefringence, Types of Birefringent crystals, Nicol Prism and its working principle.

**Methods of Polarisation – II:**

Positive and Negative uniaxial birefringent crystals, Ordinary and Extra Ordinary Rays in Birefringent Crystals, Applications of Nicol Prism

**Retarding Plates**

Theory of Retarding Plates,

Half wave and Quarter wave Plates,

Production of Elliptically and Circularly polarized light

**Following are the Courses and Syllabus I plan to offer as a part of PMRF teaching duty. I shall be providing the lecture notes, problems as well as the solutions.**

## Quantum Physics

__The Need for a New Theory:__

Blackbody Radiation, Photoelectric Effect, Compton’s Experiment, Low Temperature Specific Heat Capacity of Solids, Stern Gerlach Experiment, Young’s Double slit experiment

** Objective: **To motivate the necessity of a theory

**which can**

**explain the classically unexplainable experimental results.**

__The Shr”odinger’s Equation:__

Time-independent Schrodinger equation by Separation of Variables, Infinite Square-well, Scattering and Tunneling from 1-D potentials, Finite square-well, Linear Harmonic Oscillator, 3D-infinite square potential, Hydrogen-atom

** Objective:** To inculcate mathematical ability to find the energy eigenvalues and eigenstates for different potentials.

To develop the skill of interpretation of the mathematical results obtained from the quantum formalism

__Dirac’s Formalism:__

Ket and Bra Notation, Theory of Spin and Angular Momentum, Ladder Operator Formalism for Harmonic Oscillator, Schrodinger equation in the context of Dirac’s Formalism,

** Objective: **To develop the ability to change eigen bases(particularly, real and momentum space) as per the need of problem. To appreciate that the ladder operator formalism can bypass solving the Schrodinger equation. Motivate the power of the formalism to be applicable in advanced theories like Quantum Field Theory.

** **

__Approximation Techniques:__

Time-Independent perturbation theory, Degenerate perturbation theory,

Variational Method to approximate ground state energy value

WKB approximation method, Some historical context in which they were used.

** Objective: **To equip students with techniques to handle unsolvable potentials. To develop an ability to extract information about quantum systems without actually solving the Schrodinger equation.

__Introduction to many particle system:__

Identical Particles, Pauli Exclusion Principle

** Objective: **To appreciate the insufficiency of yet learned tools to understand many-particle systems. To motivate the need for an advanced theory which can handle large number of particles.