A Mathematical Journey to Quantum Mechanics

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