A Mathematical Journey to Quantum Mechanics

ISBN-13: 9783030861001 / Angielski / Miękka / 2022

SALVATO CAPOZZIELLO

Spis treści

1 Newtonian Mechanics, Lagrangians and Hamiltonians 15

1.1 Some Words about the Priciples of Newtonian Mechanics . . . . . . . . . . . . 15

1.2 The Mechanical Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Lagrangians and Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . 21

1.4 The Mechanical Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Hamiltonians and General Hamilton’s Equations . . . . . . . . . . . . . . . . . 27

1.6 Poisson’s Brackets in Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 29

2 Can Light Be Described by Classical Mechanics? 33

2.1 Michelson-Morley Experiment and the Principles of Special Relativity . . . . . 33

2.2 Moving among Inertial Frames: Lorentz Transformations . . . . . . . . . . . . 38

2.3 Addition of Velocities: the Relativistic Formula . . . . . . . . . . . . . . . . . . 41

2.4 Einstein’s Rest Energy Formula: E=mc2 . . . . . . . . . . . . . . . . . . . . . 42

2.5 Relativistic Energy Formula: E2 = p2 c2 + m2 c4 . . . . . . . . . . . . . . . . . 44

2.6 Describing Electromagnetic Waves: Maxwell’s Equations . . . . . . . . . . . . . 44

2.7 Invariance under Lorentz Transformations and non-Invariance under Galilei’s

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Why Quantum Mechanics? 51

3.1 What Do We Think about the Nature of Matter . . . . . . . . . . . . . . . . . 51

3.2 Monochromatic Plane Waves - the One Dimensional Case . . . . . . . . . . . . 55

3.3 Young’s Double Split Experiment: Light Seen as a Wave . . . . . . . . . . . . . 60

3.4 The Plank-Einstein formula: E=hf . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Light Seen as a Corpuscle: Einstein’s Photoelectric Eect . . . . . . . . . . . . 69

3.6 Atomic Spectra and Bohr’s Model of Hydrogen Atom . . . . . . . . . . . . . . . 70

3.7 Louis de Broglie Hypothesis: Material Objects Exhibit Wave-like Behavior . . . 73

3.8 Strengthening Einstein’s Idea: The Compton Eect . . . . . . . . . . . . . . . . 75

4 Schrödinger’s Equations and Consequences 79

4.1 The Schrödinger’s Equations - the one Dimensional Case . . . . . . . . . . . . . 79

4.2 Solving Schrödinger Equation for the Free Particle . . . . . . . . . . . . . . . . 81

4.3 Solving Schrödinger Equation for a Particle in a Box . . . . . . . . . . . . . . . 82

4.4 Solving Schrödinger Equation in the Case of Harmonic Oscillator. 

The Quantified Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 The Mathematics behind the Harmonic Oscillator 91

5.1 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Real and Complex Vector Structures . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.1 Finite Dimensional Real and Complex Vector Spaces, Inner Product,

Norm, Distance, Completeness . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.2 Pre-Hilbert and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.3 Examples of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.4 Orthogonal and Orthonormal Systems in Hilbert Spaces . . . . . . . . . 109

5.2.5 Linear Operators, Eigenvalues, Eigenvectors and Schrödinger Equation . 110

5.3 Again about de Broglie Hypothesis: Wave-Particle Duality and Wave Packets . 115

5.4 More about Electron in an Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6 Understanding Heisenberg’s Uncertainty Principle and the Mathematics

behind 121

6.1 Wave Packets and Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . 121

6.2 Wave Functions with Determined Momentum and Energy. Schrödinger’s Equation

for related Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.3 Gauss’ Wave Packet and Heisenberg Uncertainty Principle . . . . . . . . . . . . 125

6.4 The Mathematics behind the Wave Packets: Fourier Series and Fourier Transforms

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 Evolving to Quantum Mechanics Principles 143

7.1 Operators in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2 The Conservation Law . . . . . . . . . . . . . . 149

7.3 Similarities with Hamiltonian Formalism of Classical Mechanics . . . . . . . . 153

7.4 (t; x) from a Wave Function to a Quantum State of a System. The Postulates

of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8 Consequences of Quantum Mechanics Postulates 167

8.1 Ehrenfest’s Theorem and Consequences . . . . . . . . . . . . . . . . . . . . . . 167

8.2 A Consequence of QM Postulates: Heisenberg’s General Uncertainty Principle . 170

8.3 Dirac Notation and what a QM Experiment Is . . . . . . . . . . . . . . . . . . . 175

8.4 Polarization of Photons in Dirac Notation . . . . . . . . . . . . . . . . . . . . . 178

8.5 Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.6 Revisiting the Harmonic Oscillator: the Ladder Operators . . . . . . . . . . . . 197

8.7 Angular Momentum Operators in Quantum Mechanics . . . . . . . . . . . . . . 205

8.8 Gradient and Laplace Operator in Spherical Coordinates. Revisiting the Schrödinger

Equation, now in Spherical Coordinates. Legendre’s Polynomials and the Spherical

Harmonics. The Hydrogen Atom and Quantum Numbers . . . . . . . . . . 211

8.9 Pauli Matrices and Dirac Equation. Relativistic Quantum Mechanics . . . . . . 228

O autorze

Wladimir-Georges Boskoff graduated at Faculty of Mathematics of the University of Bucharest in 1982 - PhD in 1994. Since 1990, he became Member of the Department of Mathematics and Informatics of Ovidius University of Constanta providing courses in Euclidean Geometry, Differential Geometry, Calculus on Manifolds, Mechanics and Relativity, Astronomy, History of Mathematics, Basic Quantum Mechanics, etc. Among his previous books, "A Mathematical Journey to Relativity" with Salvatore Capozziello, Springer, 2020, and "Discovering Geometry: An Axiomatic Approach" with Adrian Vijiac, Matrixrom, 2011/2014.

Salvatore Capozziello is Full Professor in Astronomy and Astrophysics at the Department of Physics of University of Naples "Federico II" and Former President of the Italian Society for General Relativity and Gravitation (SIGRAV). Since 2013, he is Professor Honoris Causa at the Tomsk State Pedagogical University (TSPU), Russian Federation. His scientific activity is devoted to research topics in general relativity, cosmology, relativistic astrophysics, and physics of gravitation in their theoretical and phenomenological aspects. His research interests are extended theories of gravity and their cosmological and astrophysical applications; large-scale structure of the universe; gravitational lensing; gravitational waves; galactic dynamics; quantum phenomena in a gravitational field; quantum cosmology. He published almost 600 scientific papers and 5 books.

Blurb

This book provides an itinerary to quantum mechanics taking into account the basic mathematics to formulate it. Specifically, it features the main experiments and postulates of quantum mechanics pointing out their mathematical prominent aspects showing how physical concepts and mathematical tools are deeply intertwined. 

The material covers topics such as analytic mechanics in Newtonian, Lagrangian, and Hamiltonian formulations, theory of light as formulated in special relativity, and then why quantum mechanics is necessary to explain experiments like the double-split, atomic spectra, and photoelectric effect. The Schrödinger equation and its solutions are developed in detail. It is pointed out that, starting from the concept of the harmonic oscillator, it is possible to develop advanced quantum mechanics. Furthermore, the mathematics behind the Heisenberg uncertainty principle is constructed towards advanced quantum mechanical principles. Relativistic quantum mechanics is finally considered.

The book is devoted to undergraduate students from University courses of Physics, Mathematics, Chemistry, and Engineering. It consists of 50 self-contained lectures, and any statement and theorem are demonstrated in detail. It is the companion book of "A Mathematical Journey to Relativity", by the same Authors, published by Springer in 2020.