A Mathematical Journey to Quantum Mechanics (UNITEXT for Physics)

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 Schrdinger 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. 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 Schrdinger’s Equations and Consequences 79 4.1 The Schrdinger’s Equations – the one Dimensional Case . . . . . . . . . . . . . 79 4.2 Solving Schrdinger Equation for the Free Particle . . . . . . . . . . . . . . . . 81 4.3 Solving Schrdinger Equation for a Particle in a Box . . . . . . . . . . . . . . . 82 4.4 Solving Schrdinger 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 Schrdinger 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 Schrdinger Equation . . . . . . . . . . . . . . . . . . . . . . 121 6.2 Wave Functions with Determined Momentum and Energy. Schrdinger’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 Schrdinger 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