A Mathematical Journey to Quantum Mechanics
Salvatore / Boskoff Capozziello
289 pages, parution le 27/09/2021
Résumé
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 Schroedinger's Equations and Consequences 794.1 The Schroedinger's Equations - the one Dimensional Case . . . . . . . . . . . . . 794.2 Solving Schroedinger Equation for the Free Particle . . . . . . . . . . . . . . . . 814.3 Solving Schroedinger Equation for a Particle in a Box . . . . . . . . . . . . . . . 824.4 Solving Schroedinger 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 Schroedinger 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 Principle and the Mathematicsbehind 1216.1 Wave Packets and Schroedinger Equation . . . . . . . . . . . . . . . . . . . . . . 1216.2 Wave Functions with Determined Momentum and Energy. Schroedinger's Equationfor related Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3 Gauss' Wave Packet and Heisenberg Uncertainty Principle . . . . . . . . . . . . 1256.4 The Mathematics behind the Wave Packets: Fourier Series and Fourier Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Evolving to Quantum Mechanics Principles 1437.1 Operators in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 1437.2 The Conservation Law . . . . . . . . . . . . . . 1497.3 Similarities with Hamiltonian Formalism of Classical Mechanics . . . . . . . . 1537.4 (t; x) from a Wave Function to a Quantum State of a System. The Postulatesof Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8 Consequences of Quantum Mechanics Postulates 1678.1 Ehrenfest's Theorem and Consequences . . . . . . . . . . . . . . . . . . . . . . 1678.2 A Consequence of QM Postulates: Heisenberg's General Uncertainty Principle . 1708.3 Dirac Notation and what a QM Experiment Is . . . . . . . . . . . . . . . . . . . 1758.4 Polarization of Photons in Dirac Notation . . . . . . . . . . . . . . . . . . . . . 1788.5 Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.6 Revisiting the Harmonic Oscillator: the Ladder Operators . . . . . . . . . . . . 1978.7 Angular Momentum Operators in Quantum Mechanics . . . . . . . . . . . . . . 2058.8 Gradient and Laplace Operator in Spherical Coordinates. Revisiting the SchroedingerEquation, now in Spherical Coordinates. Legendre's Polynomials and the SphericalHarmonics. The Hydrogen Atom and Quantum Numbers . . . . . . . . . . 2118.9 Pauli Matrices and Dirac Equation. Relativistic Quantum Mechanics . . . . . . 228
Salvatore Capozziello is Full Professor in Astronomy and Astrophysics at the Department of Physics of University of Naples "Federico II" and Former President of the Italian Society for General Relativity and Gravitation (SIGRAV). Since 2013, he is Professor Honoris Causa at the Tomsk State Pedagogical University (TSPU), Russian Federation. His scientific activity is devoted to research topics in general relativity, cosmology, relativistic astrophysics, and physics of gravitation in their theoretical and phenomenological aspects. His research interests are extended theories of gravity and their cosmological and astrophysical applications; large-scale structure of the universe; gravitational lensing; gravitational waves; galactic dynamics; quantum phenomena in a gravitational field; quantum cosmology. He published almost 600 scientific papers and 5 books.
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Springer |
| Auteur(s) | Salvatore / Boskoff Capozziello |
| Parution | 27/09/2021 |
| Nb. de pages | 289 |
| EAN13 | 9783030860974 |
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