A Mathematical Journey to Quantum Mechanics (UNITEXT for Physics)

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