A mathematical journey to quantum mechanics
著者
書誌事項
A mathematical journey to quantum mechanics
（Unitext for physics）
Springer, c2021
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注記
Includes bibliographical references (p. 285-286) and index
内容説明・目次
内容説明
This book provides an itinerary to quantum mechanics taking into account the basic mathematics to formulate it. Specifically, it features the main experiments and postulates of quantum mechanics pointing out their mathematical prominent aspects showing how physical concepts and mathematical tools are deeply intertwined. The material covers topics such as analytic mechanics in Newtonian, Lagrangian, and Hamiltonian formulations, theory of light as formulated in special relativity, and then why quantum mechanics is necessary to explain experiments like the double-split, atomic spectra, and photoelectric effect. The Schroedinger equation and its solutions are developed in detail. It is pointed out that, starting from the concept of the harmonic oscillator, it is possible to develop advanced quantum mechanics. Furthermore, the mathematics behind the Heisenberg uncertainty principle is constructed towards advanced quantum mechanical principles. Relativistic quantum mechanics is finally considered.The book is devoted to undergraduate students from University courses of Physics, Mathematics, Chemistry, and Engineering. It consists of 50 self-contained lectures, and any statement and theorem are demonstrated in detail. It is the companion book of "A Mathematical Journey to Relativity", by the same Authors, published by Springer in 2020.
目次
- 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 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
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