Quantum Mechanics

The following sections are included:

• Dedication

• Preface

• Contents

The following sections are included:

• Introduction

• Breakdown of Classical Concepts

• Problems

The following sections are included:

• Uncertainty Principle

• Illustration of Heisenberg Uncertainty Principle

• Schrödinger Equation

• Physical Interpretation of Wave Function

• Stationary States

• Eigenvalues and Eigenfunctions

• Problems

The following sections are included:

• Basic Postulates of Quantum Mechanics

• Formal Properties of Quantum Mechanical Operators

• Continuous Spectrum and Dirac Delta Functions

• Uncertainty Principle and Non-Commutativity of Observables

• Problems

The following sections are included:

• One Dimensional Potential Step

• The Potential Barrier

• Parity

• An Example of Bound State (Discrete Energy Levels)

• Bound State (Square Well Potential)

• Problems

The following sections are included:

• Introduction

• Quantum Theory of Simple Harmonic Oscillator

• Properties of Hermite Polynomial

• The Average Values

• Problems

The following sections are included:

• Introduction

• Properties of Angular Momentum

• Eigenfunctions and Eigenvalues of Angular Momentum

• Properties of Associated Legendre Polynomials

• Parity of a State

• Problems

The following sections are included:

• Introduction

• Coulomb Potential

• Wave Functions of Hydrogen-like Atom

• Problems

The following sections are included:

• Two Body Problem: (Centre of Mass Motion)

• Collision Theory

• Kinematices of a Scattering Process

• Scattering Cross Section

• Scattering by a Spherical Symmetry Potential

• Phase Shifts

• Optical Theorem

• Resonances, Dispersion Formula

• Bound States and Resonances: Square Well Potential

• δ-function Potential

• The Born Approximation

• Quasi-Classical Approximation

• Problems

The following sections are included:

• State Vectors

• Schrödinger Representation

• Relation between the Momentum and Schrödinger Representations

• Matrix Mechanics

• Simple Harmonic Oscillator

• Supersymmetric Oscillator

• Problems

The following sections are included:

• Introduction

• Heisenberg Equation of Motion

• Free Particle Propagator

• Unitary Transformation

• Invariance Principles, Conservation Laws

• Discrete Transformation

• The Path Integration Formulation of Quantum Mechanics
  • Introduction
  • Path Integral Method
  • Free Particle Propagator from Path Integral Formulism
  • Motion with a Source: Generating Functional for Time-ordered Products of Heisenberg Operators

• Problems

The following sections are included:

• The Zeeman Effect

• Angular Momentum

• Matrix Representation of Angular Momentum

• Spin

• Splitting of the Ground State of Hydrogen Atom

• Addition of Spin

• Addition of Angular Momenta

• Rotations: Rotation Matrices

• Vector and Tensor Operators

• Wigner–Eckart Theorem

• Application of Wigner–Eckart Theorem

• Problems

The following sections are included:

• Introduction

• Stationary Perturbation Theory

• Degeneracy

• The Stark Effect

• Non-Perturbative Method (Variational Principle)

• Problems

The following sections are included:

• Transition Probability

• Problems

The following sections are included:

• Introduction

• Permutation Operator and Exclusion Principle

• Non-interacting Particles and Exclusion Principle

• Two Electrons System (Helium Atom)

• Scattering of Identical Particles

• Problems

The following sections are included:

• Introduction

• Magnetic Spin Resonance

• Particle Mixing
  • Introduction
  • General Formalism for Particle Mixing
  • Complex
  • Complex (q = d, s)
  • Neutrino Oscillations

• Problems

The following sections are included:

• Introduction

• Qubit

• Multiple Qubit Systems

• Matrix Representation of Qubits

• Single Qubit Logic Gates

• Quantum Logic Gates at Work

• Two Qubit Logic Gates

• Quantum Circuits

• EPR Entanglement

• Problems

The following sections are included:

• Interaction of Electromagnetic Field with Electrons

• Gauge Principle: Aharanov–Bohm Effect

• Landau Levels

• Quantization of Radiation Field

• Perturbation Induced by Electromagnetic Field

• Stimulated Emission and Absorption of Radiation

• Spontaneous Emission of Radiation: Dipole Approximation

• Decay Width
  • E₁ Transition
  • Magnetic Dipole Transition due to Intrinsic Spin

• Dipole Sum Rules, Two Point Function

• Application of Dipole Sum Rules

• Dispersion Relation

• Problems

The following sections are included:

• Introduction

• Interaction Picture

• Formal Solution of Schrödinger Equation in the Interaction Picture

• Scattering States — The S-Matrix

• The Scattering Cross-section

• Properties of the Scattering States

• The Coordinate Representation

• Unitarity of the S-Matrix

• Optical Theorem

• Problems

The following sections are included:

• Introduction

• Rotational Invariance

• Parity

• Time Reversal

• Problems

The following sections are included:

• A Brief Review of the Theory of Relativity

• Relativistic Quantum Mechanics: Introduction

• Dirac Equation

• Covariant Form of Dirac Equation

• Relativistic Invariance of Dirac Equation

• Transformation Properties of Dirac Bilinears

• Discrete Transformations

• Generators of the Lorentz Group and their Representations
  • Matrix Representation of the Generators of the Lorentz Group
  • Representations of Lorentz Group

• Gauge Invariance: Dirac Equation in the Presence of Electromagnetic Field

• Constants of Motion

• ‘Velocity’ in Dirac Theory

• Two Component Pauli Form of Dirac Equation: Existence of Magnetic Moment

• Dirac Equation for the Hydrogen-like Atom

• Hole Theory and Existence of Positrons (Anti-matter)

• Free Particle Solutions, Plane Wave Solutions of Dirac Equation

• Charge Conjugation

• Large and Small Bilinear Covariants

• Weyl Equation

• Problems

The following sections are included:

• Dirac Equation in (1+2) Dimensions
  • Dirac Equation in Hamiltonian Form and Plane Wave Solutions

• Dirac Equation And Graphene
  • Introduction
  • Electronic Properties of Graphene
  • Cyclotron Mass
  • Dirac Hamiltonian and Dirac Equation for Quasiparticles in Graphene
  • Tunneling and Klein Paradox
  • Dirac Fermions in an External Gauge Field (Electromagnetic): Landau Levels

• Problems

• Selected References and Further Reading

The following sections are included:

• Index