Building Blocks of Quantum Mechanics: Theory and Applications

Description

This textbook provides a concise yet comprehensive introduction to the principles, concepts, and methods of quantum mechanics. It covers the basic building blocks of quantum mechanics theory and applications, illuminated throughout by physical insights and examples of quantum mechanics, such as the one-dimensional eigen-problem, the harmonic oscillator, the Aharonov-Bohm effect, Landau levels, the hydrogen atom, the Landau-Zener transition and the Berry phase. This self-contained textbook is suitable for junior and senior undergraduate students, in addition to advanced students who have studied general physics (including classical mechanics, electromagnetics, and atomic physics), calculus, and linear algebra. Key features: Presents an accessible and concise treatment of quantum mechanics Contains a wealth of case studies and examples to illustrate concepts Based off the author's established course and lecture notes Cover Half Title Title Page Copyright Page Dedication Contents Preface Notations Formulas in SI units and Gaussian units Table of fundamental constants Chapter 1: Introduction 1.1. Brief history of quantum mechanics 1.2. Schrödinger equation 1.3. Probability interpretation of wave function 1.4. Stationary Schrödinger equation 1.5. Conservation of probability 1.6. Quantum superposition 1.6.1. No cloning theorem 1.6.2. Schrödinger cat 1.7. Operators 1.8. Quantum measurement 1.8.1. Stern-Gerlach experiment 1.9. Expectation values 1.10. Problems Chapter 2: One-dimensional Eigen-problem 2.1. Symmetric potential and parity 2.2. Free particle 2.3. Delta-function normalization 2.4. Infinite square well potential 2.5. Finite square well potential 2.5.1. Bound states −V 0 2.6. Quantum tunneling 2.7. Delta-function potential 2.7.1. Bound state (α < 0 and E 0) 2.8. The WKB approximation 2.8.1. Solution around a turning point 2.8.2. The connection formulae 2.8.3. Quantization of energy levels 2.9. Problems Chapter 3: Representation theory of quantum states 3.1. Representation 3.1.1. Dirac bracket notations 3.1.2. Representation of quantum states 3.1.3. Hermitian operators 3.1.4. Eigenstates of Hermitian operators 3.1.5. Representation of operators 3.1.6. Representation of Schrödinger equation 3.1.7. Feynman-Hellmann theorem 3.2. Basis transformation 3.2.1. Example: From real to momentum space representation 3.3. Commutators 3.3.1. Properties of commutable operators 3.3.2. Properties of noncommutable operators 3.4. Schrödinger picture 3.4.1. Virial theorem 3.4.2. Ehrenfest theorem 3.5. Heisenberg picture 3.6. Uncertainty principle 3.7. The time-energy uncertainty principle 3.8. Problems Chapter 4: Harmonic Oscillators 4.1. One-dimensional harmonic oscillator 4.1.1. Ladder operators 4.1.2. Eigen-spectrum 4.1.3. Eigenfunction 4.1.4. Occupation representation 4.2. Coherent state 4.2.1. Minimum uncertainty state 4.2.2. Wave function of the coherent state 4.3. Charged particles in an electromagnetic field 4.3.1. Minimal coupling 4.3.2. Gauge invariance 4.3.3. Probability current 4.3.4. Aharonov-Bohm effect 4.4. Landau levels 4.4.1. Landau gauge 4.4.2. Degeneracy of Landau levels 4.4.3. Symmetric gauge 4.4.4. Lowest Landau level 4.5. Problems Chapter 5: Angular Momentum 5.1. Orbital angular momentum 5.2. General angular momentum 5.2.1. Matrix representation of angular momentum operators 5.3. Eigenfunctions of orbital angular momentum 5.4. Spin angular momentum 5.4.1. Pauli matrices 5.4.2. Eigenstates of S = 1/2 5.4.3. Qubit and Bloch sphere 5.5. Addition of two angular momenta 5.5.1. Clebsch-Gordan coefficients 5.5.2. Addition of two S = 1/2. spins 5.6. Wigner-Eckart theorem∗ 5.6.1. Proof of the Wigner-Eckart theorem 5.7. Problems Chapter 6: Central potential 6.1. Three-dimensional potential with spherical symmetry 6.2. Hydrogenic atom 6.2.1. Hamiltonian in the center-of-mass framework 6.2.2. Bound state solutions 6.2.3. Solutions by series expansion 6.2.4. Radial wave function 6.2.5. Rydberg formula 6.3. Partial wave method 6.3.1. Partial wave expansion 6.3.2. Scattering amplitude 6.3.3. Scattering cross section 6.3.4. Hard-sphere scattering 6.4. Supersymmetric quantum mechanics approach∗ 6.4.1. Supersymmetric solution of the hydrogenic atom 6.5. Problems Chapter 7: Identical Particles 7.1. Permutation symmetry 7.2. Bose-Einstein and Fermi-Dirac statistics 7.2.1. Exchange degeneracy 7.2.2. Anti-symmetrized wave functions 7.2.3. Symmetrized wave functions 7.2.4. Two identical particles 7.2.5. Exchange force 7.3. Free fermion gas 7.3.1. Particle in a periodic box 7.3.2. Fermi surface 7.3.3. Degeneracy pressure 7.4. Hydrogen molecule 7.5. Entanglement 7.5.1. Density matrix 7.5.2. Entangled state 7.5.3. Entanglement entropy 7.5.4. Bell bases 7.5.5. EPR paradox 7.6. Problems Chapter 8: Symmetry and Conservation Law 8.1. Spatial translation invariance and momentum conservation 8.1.1. Translation operator 8.1.2. Generator of translations 8.1.3. Momentum conservation 8.1.4. Finite translation 8.1.5. Stone theorem 8.2. Galilean invariance 8.3. Noether theorem 8.4. Rotation and angular momentum conservation 8.4.1. Rotation in two dimensions 8.4.2. Rotation in three dimensions 8.4.3. Angular momentum conservation 8.5. Time translation invariance and energy conservation 8.6. Time-reversal symmetry 8.6.1. Time-reversal transformation 8.6.2. Even and odd operators 8.6.3. Nondegenerate energy eigenstate 8.6.4. Kramers degeneracy 8.7. Problems Chapter 9: Approximate methods 9.1. Ground state wave function is node free 9.2. Variational ansatz 9.2.1. Half-harmonic oscillator 9.2.2. Ground state of helium 9.3. Stationary perturbation theory 9.3.1. First order correction 9.3.2. Second order correction 9.3.3. Anharmonic oscillator 9.4. Degenerate perturbation theory 9.4.1. Linear Stark effect 9.5. Problems Chapter 10: Quantum transition 10.1. Time-dependent Schr¨odinger equation 10.1.1. Perturbation expansion 10.1.2. First order correction 10.1.3. Two-level systems 10.2. Monochromatic perturbation 10.2.1. Interaction of atoms with electromagnetic wave 10.2.2. Absorption and stimulated emission 10.2.3. Fermi’s golden rule 10.2.4. Selection rules∗ 10.2.5. Constant perturbation 10.3. Einstein’s theory of radiation 10.4. Problems Chapter 11: Adiabatic and diabatic Evolution 11.1. Adiabatic versus sudden approximations 11.1.1. The adiabatic theorem 11.1.2. Sudden approximation 11.1.3. Quantum Zeno effect 11.2. Landau-Zener transition 11.2.1. Rabi oscillation 11.2.2. Landau-Zener model 11.2.3. Landau-Zener transition 11.2.4. Derivation of the Landau-Zener formulas∗ 11.3. Berry’s phase 11.3.1. Fictitious gauge vector 11.3.2. Fictitious magnetic field 11.3.3. Quantization of fictitious magnetic flux 11.3.4. A spin-1/2. particle in a magnetic flux 11.3.5. Charged particle moving around a magnetic flux 11.4. Problems Chapter 12: Relativistic Quantum Mechanics 12.1. Relativistic covariance 12.2. Klein-Gordon equation 12.2.1. Current and density operators 12.2.2. Interpretation of the density operator 12.2.3. Negative energy 12.2.4. Nonrelativistic limit 12.3. Dirac equation 12.3.1. “Derivation” of the Dirac equation 12.3.2. Charge conservation 12.3.3. Covariant form and gamma matrices 12.3.4. Coupled with electromagnetic fields 12.3.5. Free-particle solutions 12.3.6. Spin: rotational symmetry 12.3.7. Antiparticles: charge conjugate 12.3.8. Negative energy: Dirac sea 12.4. Nonrelativistic limit of the Dirac equation 12.4.1. First order approximation 12.4.2. Second order approximation 12.4.3. Normalization 12.4.4. Effective Hamiltonian 12.5. Problems Index