Luca Salasnich is Full Professor of Condensed Matter Theory at the Department of Physics and Astronomy "Galileo Galilei," University of Padova, Italy. He was awarded an M.Sc. in Physics by the University of Padova in 1991, and his Ph.D. in Theoretical Physics by the University of Florence in 1995. His fields of research are condensed matter theory and statistical physics, in particular nonlinear phenomena and macroscopic quantum effects (like superfluidity and superconductivity) in ultra-cold atomic gases and other many-body systems. He has written more than 200 peer-reviewed scientific papers in international journals, with over 4700 citations.

Table of 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

2 Special Relativity

2.1 Lorentz Transformations

2.2 Einstein Postulates

2.2.1 Gedanken Experiment of Einstein

2.3 Relativistic Mechanics

2.3.1 Relativistic Kinematics

2.3.2 Relativistic Dynamics

3 Quantum Properties of Light

3.1 Black-Body Radiation

3.1.1 Ideal Black Body

3.1.2 Derivation of the Planck Law

3.2 Photoelectric E ect

3.2.1 Experimental Data

3.2.2 Theoretical Explanation

3.3 Energy and Linear Momentum of a Photon

3.4 Compton E ect

3.4.1 Theoretical Explanation

4 Quantum Energy Levels of Atoms

4.1 Energy Spectra

4.1.1 Energy Spectrum of Hydrogen Atom

4.2 Hydrogen Atom of Bohr

4.2.1 Derivation of the Bohr Results

4.3 Energy Levels and Photons

4.4 Electromagnetic Transitions

5 Wave Properties of Matter

5.1 De Broglie Wavelength

5.1.1 Explaining the Bohr Quantization

5.2 Experiment of Davisson and Germer

5.3 Double-Slit Experiment with Light

5.4 Double-Slit Experiment with Electrons

5.6 Matrix Quantum Mechanics of Heisenberg, Born and Jordan

5.7 Wave Quantum Mechanics of Schrodinger

5.7.1 Derivation of the Schr odinger Equation

5.8 Formal Quantization Rules

5.8.1 Schrodinger Equation for a Free Particle

5.8.2 Schrodinger Equation for a Particle in an External Potential

5.9 Stationary Schr odinger Equation

6 Axioms of Quantum Mechanics

6.1 Matrix Mechanics

6.2 Axioms of Quantum Mechanics

7 Applications of Quantum Mechanics

7.1 Quantum Particle in a One-Dimensional Box Potential

7.2 Quantum Particle in a One-Dimensional Harmonic Potential

8 Quantum Physics of Atoms

8.1 Quantum Particle in a Separable Potential

8.1.1 Quantum Particle in the Harmonic Potential

8.2 Dirac Notation for a Quantum State

8.3 Electron in the Hydrogen Atom

8.3.1 Schrodinger Equation in Spherical Polar Coordinates

8.3.2 Selection Rules

8.4 Pauli Exclusion Principle and the Spin

8.5 Semi-Integer and Integer Spin: Fermions and Bosons

8.6 The Dirac Equation

8.6.1 The Pauli Equation and the Spin

8.6.2 Dirac Equation with a Central Potential

8.6.3 Relativistic Hydrogen Atom and Fine Splitting

8.6.4 Relativistic Corrections to the Schrodinger Hamiltonian

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 Variational Principle

9.3.2 Electrons in Atoms and Molecules

10 Quantum Statistical Mechanics

10.1 Quantum Statistical Ensembles

10.1.1 Quantum Microcanonical 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

A.1 The Heaviside Step Function

A.2 The Strange Function of Dirac

A.2.1 Dirac Function and the Integrals

A.3 Dirac Function in D Spatial Dimensions

Appendix B Complex Numbers

B.1 Set of Complex Numbers

B.2.1 Polar Representation

B.3 Euler Formula

B.3.1 Proof of the Euler Formula

B.3.2 De Moivre Formula

B.4 Fundamental Theorem of Algebra

B.5 Complex Functions

Appendix C Fourier Transform

C.1 Geometric and Taylor Series

C.2 Fourier Series .

C.2.1 Complex Representation of the Fourier Series

C.3 Fourier Integral

C.3.1 Properties of the Fourier Transform

C.3.2 Fourier Transform and Uncertanty Theorem

Appendix D Di erential equations

D.1 First-Order Ordinary Di erential Equations

D.1.1 Separation of Variables

D.2 Second-Order Ordinary Di erential Equations

D.3 Newton Law as a Second-Order ODE

D.4 Partial Di erential Equations

D.4.1 Wave Equation

D.4.2 Di usion Equation

Bibliography