This book offers an introduction to statistical mechanics, special relativity, and quantum physics. It is based on the lecture notes prepared for the one-semester course of "Quantum Physics" belonging to the Bachelor of Science in Material Sciences at the University of Padova.
The first chapter briefly reviews the ideas of classical statistical mechanics introduced by James Clerk Maxwell, Ludwig Boltzmann, Willard Gibbs, and others. The second chapter is devoted to the special relativity of Albert Einstein. In the third chapter, it is historically analyzed the quantization of light due to Max Planck and Albert Einstein, while the fourth chapter discusses the Niels Bohr quantization of the energy levels and the electromagnetic transitions. The fifth chapter investigates the Schrodinger equation, which was obtained by Erwin Schrodinger from the idea of Louis De Broglie to associate to each particle a quantum wavelength. Chapter Six describes the basic axioms of quantum mechanics, which were formulated in the seminal books of Paul Dirac and John von Neumann. In chapter seven, there are several important application of quantum mechanics: the quantum particle in a box, the quantum particle in the harmonic potential, the quantum tunneling, the stationary perturbation theory, and the time-dependent perturbation theory. Chapter Eight is devoted to the study of quantum atomic physics with special emphasis on the spin of the electron, which needs the Dirac equation for a rigorous theoretical justification. In the ninth chapter, it is explained the quantum mechanics of many identical particles at zero temperature, while in Chapter Ten the discussion is extended to many quantum particles at finite temperature by introducing and using the quantum statistical mechanics.
Author(s): Luca Salasnich
Series: UNITEXT for Physics
Edition: 1
Publisher: Springer Nature Switzerland AG
Year: 2022
Language: English
Pages: 194
City: Cham, Switzerland
Tags: Statistical Mechanics, Relativity, Quantum Physics, Light, Matter, Atoms
Preface
Contents
1 Classical Statistical Mechanics
1.1 Kinetic Theory of Gases
1.1.1 Maxwell Distribution of Velocities
1.1.2 Maxwell-Boltzmann Distribution of Energies
1.1.3 Single-Particle Density of States
1.2 Statistical Ensembles of Gibbs
1.2.1 Microcanonical Ensemble
1.2.2 Canonical Ensemble
1.2.3 Grand Canonical Ensemble
1.2.4 Many-Particle Density of States
1.3 Heat Capacity of Gases and Solids
2 Special and General Relativity
2.1 Electromagnetic Waves
2.1.1 Lorentz Invariance of d'Alembert Operator
2.2 Lorentz Transformations
2.2.1 Thought Experiment with Light Bulb
2.3 Einstein Postulates
2.4 Relativistic Kinematics
2.4.1 Length Contraction
2.4.2 Time Dilation
2.4.3 Transformation of Velocities
2.5 Relativistic Dynamics
2.5.1 Mechanical Work and Relativistic Energy
2.5.2 Relativistic Energy and Linear Momentum
2.5.3 Non-relativistic Limit of the Energy
2.6 Basic Concepts of General Relativity
2.6.1 Spacetime Interval
2.6.2 Curved Manifolds
2.6.3 Equivalence Principle and Einstein Equations
2.6.4 Non-Relativistic Limit of General Relativity
2.6.5 Predictions of General Relativity
3 Quantum Properties of Light
3.1 Black-Body Radiation
3.1.1 Derivation of Planck's Law
3.2 Photoelectric Effect
3.2.1 Theoretical Explanation
3.3 Energy and Linear Momentum of a Photon
3.4 Compton Effect
3.4.1 Theoretical Explanation
3.5 Pair Production
4 Quantum Properties of Matter
4.1 Heat Capacity of Solids: Einstein Versus Debye
4.2 Energy Spectra of Atoms
4.2.1 Energy Spectrum of Hydrogen Atom
4.3 Bohr's Model of Hydrogen Atom
4.3.1 Derivation of Bohr's Formula
4.4 Energy Levels and Photons
4.5 Electromagnetic Transitions
4.6 Einstein Coefficients
4.7 Life-Time of an Atomic State
4.8 Natural Line Width
4.8.1 Collisional Broadening
4.8.2 Doppler Broadening
4.9 Old Quantum Mechanics of Bohr, Wilson and Sommerfeld
5 Wavefunction of a Quantum Particle
5.1 De Broglie Wavelength
5.1.1 Explaining the Bohr Quantization
5.2 Wave Mechanics of Schrödinger
5.2.1 Derivation of Schrödinger's Equation
5.3 Double-Slit Experiment with Electrons
5.4 Formal Quantization Rules
5.4.1 Schrödinger Equation for a Free Particle
5.4.2 Schrödinger Equation for a Particle in an External Potential
5.5 Madelung Transformation
5.6 Stationary Schrödinger Equation
5.6.1 Properties of the Hamiltonian Operator
5.6.2 Orthogonality of Eigenfunctions
6 Axiomatization of Quantum Mechanics
6.1 Matrix Mechanics and Commutation Rules
6.1.1 Momentum Representation
6.2 Time Evolution Operator
6.3 Axioms of Quantum Mechanics
6.4 Heisenberg Uncertainty Principle
6.4.1 Uncertainty Principle for Non-commuting Operators
6.5 Time-Independent Perturbation Theory
6.6 Time-Dependent Perturbation Theory and Fermi Golden Rule
6.7 Variational Principle
7 Solvable Problems in Quantum Mechanics
7.1 One-Dimensional Square-Well Potential
7.2 One-Dimensional Harmonic Potential
7.2.1 Properties of Number Operator
7.3 One-Dimensional Scattering
7.4 One-Dimensional Double-Well Potential
7.4.1 One-Dimensional Double-Square-Well Potential
7.5 WKB Method
7.6 Three-Dimensional Separable Potential
7.6.1 Three-Dimensional Harmonic Potential
8 Modern Quantum Physics of Atoms
8.1 Electron in the Hydrogen Atom
8.1.1 Schrödinger Equation in Spherical Polar Coordinates
8.1.2 Selection Rules
8.2 Pauli Exclusion Principle and the Spin
8.2.1 Semi-integer and Integer Spin: Fermions and Bosons
8.3 The Dirac Equation
8.3.1 The Pauli Equation and the Spin
8.3.2 Dirac Equation with a Central Potential
8.3.3 Relativistic Hydrogen Atom and Fine Splitting
8.3.4 Relativistic Corrections to the Schrödinger Hamiltonian
8.4 Spin Properties in a Magnetic Field
8.5 Stark Effect
8.6 Zeeman Effect
8.6.1 Strong-Field Zeeman Effect
8.6.2 Weak-Field Zeeman Effect
9 Quantum Mechanics of Many-Body Systems
9.1 Identical Quantum Particles
9.1.1 Spin-Statistics Theorem
9.2 Non-interacting Identical Particles
9.2.1 Atomic Shell Structure and the Periodic Table of the Elements
9.3 Interacting Identical Particles
9.3.1 Electrons in Atoms and Molecules
9.4 The Hartree-Fock Method
9.4.1 Hartree for Bosons
9.4.2 Hartree-Fock for Fermions
10 Quantum Statistical Mechanics
10.1 Quantum Statistical Ensembles
10.1.1 Quantum Microcanonical Ensemble
10.1.2 Quantum Canonical Ensemble
10.1.3 Quantum Grand Canonical Ensemble
10.2 Bosons and Fermions at Finite Temperature
10.2.1 Gas of Photons at Thermal Equlibrium
10.2.2 Gas of Massive Bosons at Thermal Equlibrium
10.2.3 Gas of Non-interacting Fermions at Zero Temperature
Appendix A Dirac Delta Function
The Heaviside Step Function
The Strange Function of Dirac
Dirac Function and the Integrals
Dirac Function in D Spatial Dimensions
Appendix B Complex Numbers
Set of Complex Numbers
Gauss Plane
Polar Representation
Euler Formula
Proof of the Euler Formula
De Moivre Formula
Fundamental Theorem of Algebra
Complex Functions
Appendix C Fourier Transform
Geometric and Taylor Series
Fourier Series
Complex Representation of the Fourier Series
Fourier Integral
Properties of the Fourier Transform
Fourier Transform and Uncertanty Theorem
Fourier Transform of Space-Time Functions
Appendix D Differential Equations
First-Order ODE
Separation of Variables
Second-Order ODE
Newton Law as a Second-Order ODE
Partial Differential Equations
Wave Equation
Diffusion Equation
Appendix References
Index
