A Mathematical Journey to Quantum Mechanics

Description

Preface Contents 1 Introduction: How to Read This Book 2 Newtonian, Lagrangian and Hamiltonian Mechanics 2.1 Lecture 1: A Summary of the Principles of Newtonian Mechanics 2.2 Lecture 2: The Mechanical Lagrangian 2.3 Lecture 3: The Euler–Lagrange Equations 2.4 Lecture 4: The Mechanical Hamiltonian 2.5 Lecture 5: The Hamilton Equations 2.6 Lecture 6: Poisson's Brackets in Hamiltonian Mechanics 3 Can Light Be Described by Classical Mechanics? 3.1 Lecture 7: The Michelson–Morley Experiment and the Principles of Special Relativity 3.2 Lecture 8: Motion Among Inertial Frames. The Lorentz Transformations 3.3 Lecture 9: Addition of Velocities. The Relativistic Formula 3.4 Lecture 10: The Einstein Rest Energy Formula E=m c2 3.5 Lecture 11: The Relativistic Energy Formula E2= p2 c 2 + m2 c4 3.6 Lecture 12: Electromagnetic Waves by the Maxwell Equations 3.7 Lecture 13: The Invariance of Maxwell Equations Under the Lorentz Transformations 4 Why Quantum Mechanics? 4.1 Lecture 14: The Problem of the Nature of Matter 4.2 Lecture 15: Monochromatic Plane Waves—The One Dimensional Case 4.3 Lecture 16: The Young Double Split Experiment. Light Seen as a Wave 4.4 Lecture 17: The Planck–Einstein Formula E=hν 4.5 Lecture 18: Light as Particles. The Einstein Photoelectric Effect 4.6 Lecture 19: Atomic Spectra and Bohr's Model of Hydrogen Atom 4.7 Lecture 20: De Broglie's Hypothesis. Material Objects as Waves 4.8 Lecture 21: Strengthening the Einstein Idea of Photons. The Compton Effect 5 The Schrödinger Equations and Their Consequences 5.1 Lecture 22: The Schrödinger Equations. The One Dimensional Case 5.2 Lecture 23: Solving the Schrödinger Equation for the Free Particle 5.3 Lecture 24: Solving the Schrödinger Equation for a Particle in a Box 5.4 Lecture 25: Solving the Schrödinger Equation … 6 The Mathematics Behind the Harmonic Oscillator 6.1 Lecture 26: The Hermite Polynomials 6.2 Lecture 27: Real and Complex Vector Structures 6.2.1 Finite Dimensional Real and Complex Vector Spaces, Inner Product, Norm, Distance, Completeness 6.3 Lecture 28: Pre-Hilbert and Hilbert Spaces 6.4 Lecture 29: Examples of Hilbert Spaces 6.5 Lecture 30: Orthogonal and Orthonormal Systems in Hilbert Spaces 6.6 Lecture 31: Linear Operators, Eigenvalues, Eigenvectors for the Schrödinger Equation 7 From Monochromatic Plane Waves to Wave Packets 7.1 Lecture 32: Again on the de Broglie Hypothesis. Wave-Particle Duality and Wave Packets 7.2 Lecture 33: More About Electron in an Atom 8 The Heisenberg Uncertainty Principle and the Mathematics Behind 8.1 Lecture 34: Wave Packets and the Schrödinger Equation 8.2 Lecture 35: The Wave Function Ψ Solution of the Schrödinger Equation 8.3 Lecture 36: The Gauss Wave Packet and the Heisenberg Uncertainty Principle 8.4 Lecture 37: The Mathematics Behind the Wave Packets. The Fourier … 9 The Principles of Quantum Mechanics 9.1 Lecture 38: Operators in Quantum Mechanics 9.2 Lecture 39: The Relation φ*2Ψ-Ψ2φ*=div(φ*Ψ-Ψφ*) and Its Consequences 9.3 Lecture 40. Similarities with Hamiltonian Formalism of Classical Mechanics 9.4 Lecture 41: From the Wave Function to the Quantum State. The Postulates of Quantum Mechanics 10 Consequences of Quantum Mechanics Principles 10.1 Lecture 42: The Ehrenfest Theorem 10.2 Lecture 43: The Heisenberg General Uncertainty Principle 10.3 Lecture 44: The Dirac Notation and the Meaning of a Quantum Mechanics Experiment 10.4 Lecture 45: The Photon Polarization by Dirac's Notation 11 Quantum Mechanics at the Next Level 11.1 Lecture 46: The Electron Spin 11.2 Lecture 47: Revisiting the Harmonic Oscillator. The Ladder Operators 11.3 Lecture 48: The Angular Momentum in Quantum Mechanics 11.4 Lecture 49: From Differential Operators in Spherical Coordinates to the Hydrogen Atom 11.5 Lecture 50: The Pauli Matrices and the Dirac … 12 Conclusions Appendix Bibliography Index