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.