A Mathematical Journey to Quantum Mechanics

By Salvatore Capozziello and Wladimir-Georges Boskoff

Book Overview

1 Newtonian Mechanics, Lagrangians and Hamiltonians 151.1 Some Words about the Priciples of Newtonian Mechanics . . . . . . . . . . . . 151.2 The Mechanical Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3 Lagrangians and Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . 211.4 The Mechanical Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 Hamiltonians and General Hamilton's Equations . . . . . . . . . . . . . . . . . 271.6 Poisson's Brackets in Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 29
2 Can Light Be Described by Classical Mechanics? 332.1 Michelson-Morley Experiment and the Principles of Special Relativity . . . . . 332.2 Moving among Inertial Frames: Lorentz Transformations . . . . . . . . . . . . 382.3 Addition of Velocities: the Relativistic Formula . . . . . . . . . . . . . . . . . . 412.4 Einstein's Rest Energy Formula: E=mc2 . . . . . . . . . . . . . . . . . . . . . 422.5 Relativistic Energy Formula: E2 = p2 c2 + m2 c4 . . . . . . . . . . . . . . . . . 442.6 Describing Electromagnetic Waves: Maxwell's Equations . . . . . . . . . . . . . 442.7 Invariance under Lorentz Transformations and non-Invariance under Galilei'sTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Why Quantum Mechanics? 513.1 What Do We Think about the Nature of Matter . . . . . . . . . . . . . . . . . 513.2 Monochromatic Plane Waves - the One Dimensional Case . . . . . . . . . . . . 553.3 Young's Double Split Experiment: Light Seen as a Wave . . . . . . . . . . . . . 603.4 The Plank-Einstein formula: E=hf . . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Light Seen as a Corpuscle: Einstein's Photoelectric Eect . . . . . . . . . . . . 693.6 Atomic Spectra and Bohr's Model of Hydrogen Atom . . . . . . . . . . . . . . . 703.7 Louis de Broglie Hypothesis: Material Objects Exhibit Wave-like Behavior . . . 733.8 Strengthening Einstein's Idea: The Compton Eect . . . . . . . . . . . . . . . . 75
4 Schr?dinger's Equations and Consequences 794.1 The Schr?dinger's Equations - the one Dimensional Case . . . . . . . . . . . . . 794.2 Solving Schr?dinger Equation for the Free Particle . . . . . . . . . . . . . . . . 814.3 Solving Schr?dinger Equation for a Particle in a Box . . . . . . . . . . . . . . . 824.4 Solving Schr?dinger Equation in the Case of Harmonic Oscillator. The Quantified Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 The Mathematics behind the Harmonic Oscillator 915.1 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Real and Complex Vector Structures . . . . . . . . . . . . . . . . . . . . . . . . 975.2.1 Finite Dimensional Real and Complex Vector Spaces, Inner Product, Norm, Distance, Completeness . . . . . . . . . . . . . . . . . . . . . . . 975.2.2 Pre-Hilbert and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . 1005.2.3 Examples of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2.4 Orthogonal and Orthonormal Systems in Hilbert Spaces . . . . . . . . . 1095.2.5 Linear Operators, Eigenvalues, Eigenvectors and Schr?dinger Equation . 1105.3 Again about de Broglie Hypothesis: Wave-Particle Duality and Wave Packets . 1155.4 More about Electron in an Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Understanding Heisenberg's Uncertainty Principl

• Format: Paperback
• Language: English
• ISBN: 3030861007
• ISBN13: 9783030861001
• Release Date: September 2022
• Publisher: Springer
• Length: 289 Pages
• Weight: 0.96 lbs.
• Dimensions: 0.7" x 6.1" x 9.2"