Mastering Quantum Mechanics: Essentials, Theory, and Applications

Title Page
Copyright
Dedication
Table of Contents
Preface
I: Essentials
	1. Key Features of Quantum Mechanics
		1.1. Linearity of the Equations of Motion
		1.2. Complex Numbers Are Essential
		1.3. Loss of Determinism
		1.4. Quantum Superpositions
		1.5. Entanglement
		1.6. Making Atoms Possible
		Problems
	2. Light, Particles, and Waves
		2.1. Mach-Zehnder Interferometer
		2.2. Elitzur-Vaidman Bombs
		2.3. Toward Perfect Bomb Detection
		2.4. Photoelectric Effect
		2.5. Compton Scattering
		2.6. Matter Waves
		2.7. De Broglie Wavelength and Galilean Transformations
		2.8. Stationary Phase and Group Velocity
		Problems
	3. Schrödinger Equation
		3.1. The Wave Function for a Free Particle
		3.2. Equations for a Wave Function
		3.3. Schrödinger Equation for a Particle in a Potential
		3.4. Interpreting the Wave Function
		3.5. Normalization and Time Evolution
		3.6. The Wave Function as a Probability Amplitude
		3.7. The Probability Current
		3.8. Probability Current in Three Dimensions and Current Conservation
		Problems
	4. Wave Packets, Uncertainty, and Momentum Space
		4.1. Wave Packets and Uncertainty
		4.2. Wave Packet Shape Changes
		4.3. Time Evolution of a Free Wave Packet
		4.4. Uncovering Momentum Space
		Problems
	5. Expectation Values and Hermitian Operators
		5.1. Expectation Values of Operators
		5.2. Time Dependence of Expectation Values
		5.3. Hermitian Operators and Axioms of Quantum Mechanics
		5.4. Free Particle on a Circle—a First Look
		5.5. Uncertainty
		Problems
	6. Stationary States I: Special Potentials
		6.1. Stationary States
		6.2. Solving for Energy Eigenstates
		6.3. Free Particle on a Circle—a Second Look
		6.4. The Infinite Square Well
		6.5. The Finite Square Well
		6.6. The Delta Function Potential
		6.7. The Linear Potential
		Problems
	7. Stationary States II: General Features
		7.1. General Properties
		7.2. Bound States in Slowly Varying Potentials
		7.3. Sketching Wave Function Behavior
		7.4. The Node Theorem
		7.5. Shooting Method
		7.6. Removing Units from the Schrödinger Equation
		7.7. Virial Theorem
		7.8. Variational Principle
		7.9. Hellmann-Feynman Lemma
		Problems
	8. Stationary States III: Scattering
		8.1. The Step Potential
		8.2. Wave Packets in the Step Potential
		8.3. Resonant Transmission in a Square Well
		Problems
	9. Harmonic Oscillator
		9.1. Harmonic Oscillator
		9.2. Solving the Harmonic Oscillator Differential Equation
		9.3. Algebraic Solution for the Spectrum
		9.4. Excited States of the Oscillator
		Problems
	10. Angular Momentum and Central Potentials
		10.1. Angular Momentum in Quantum Mechanics
		10.2. Schrödinger Equation in Three Dimensions and Angular Momentum
		10.3. The Angular Momentum Operator
		10.4. Commuting Operators and Rotations
		10.5. Eigenstates of Angular Momentum
		10.6. The Radial Equation
		Problems
	11. Hydrogen Atom
		11.1. The Two-Body Problem
		11.2. Hydrogen Atom: Potential and Scales
		11.3. Hydrogen Atom: Bound State Spectrum
		11.4. Rydberg Atoms
		11.5. Degeneracies and Semiclassical Electron Orbits
		Problems
	12. The Simplest Quantum System: Spin One-Half
		12.1. A System with Two States
		12.2. The Stern-Gerlach Experiment
		12.3. Spin States
		12.4. Quantum Key Distribution
		Problems
II: Theory
	13. Vector Spaces and Operators
		13.1. Vector Spaces
		13.2. Subspaces, Direct Sums, and Dimensionality
		13.3. Linear Operators
		13.4. Null Space, Range, and Inverses of Operators
		13.5. Matrix Representation of Operators
		13.6. Eigenvalues and Eigenvectors
		13.7. Functions of Linear Operators and Key Identities
		Problems
	14. Inner Products, Adjoints, and Bra-kets
		14.1. Inner Products
		14.2. Orthonormal Bases
		14.3. Orthogonal Projectors
		14.4. Linear Functionals and Adjoint Operators
		14.5. Hermitian and Unitary Operators
		14.6. Remarks on Complex Vector Spaces
		14.7. Rotation Operators for Spin States
		14.8. From Inner Products to Bra-kets
		14.9. Operators, Projectors, and Adjoints
		14.10. Nondenumerable Basis States
		Problems
	15. Uncertainty Principle and Compatible Operators
		15.1. Uncertainty Defined
		15.2. The Uncertainty Principle
		15.3. Energy-Time Uncertainty
		15.4. Lower Bounds for Ground State Energies
		15.5. Diagonalization of Operators
		15.6. The Spectral Theorem
		15.7. Simultaneous Diagonalization of Hermitian Operators
		15.8. Complete Set of Commuting Observables
		Problems
	16. Pictures of Quantum Mechanics
		16.1. Schrödinger Picture and Unitary Time Evolution
		16.2. Deriving the Schrödinger Equation
		16.3. Calculating the Time Evolution Operator
		16.4. Heisenberg Picture
		16.5. Heisenberg Equations of Motion
		16.6. Axioms of Quantum Mechanics
		Problems
	17. Dynamics of Quantum Systems
		17.1. Basics of Coherent States
		17.2. Heisenberg Picture for Coherent States
		17.3. General Coherent States
		17.4. Photon States
		17.5. Spin Precession in a Magnetic Field
		17.6. Nuclear Magnetic Resonance
		17.7. Two-State System Viewed as a Spin System
		17.8. The Factorization Method
		Problems
	18. Multiparticle States and Tensor Products
		18.1. Introduction to the Tensor Product
		18.2. Operators on the Tensor Product Space
		18.3. Inner Products for Tensor Spaces
		18.4. Matrix Representations and Traces
		18.5. Entangled States
		18.6. Bell Basis States
		18.7. Quantum Teleportation
		18.8. EPR and Bell Inequalities
		18.9. No-Cloning Property
		Problems
	19. Angular Momentum and Central Potentials: Part II
		19.1. Angular Momentum and Quantum Vector Identities
		19.2. Properties of Angular Momentum
		19.3. Multiplets of Angular Momentum
		19.4. Central Potentials and Radial Equation
		19.5. Free Particle and Spherical Waves
		19.6. Rayleigh’s Formula
		19.7. The Three-Dimensional Isotropic Oscillator
		19.8. The Runge-Lenz Vector
		Problems
	20. Addition of Angular Momentum
		20.1. Adding Apples to Oranges?
		20.2. Adding Two Spin One-Half Angular Momenta
		20.3. A Primer in Perturbation Theory
		20.4. Hyperfine Splitting
		20.5. Computation of
		20.6. Spin-Orbit Coupling
		20.7. General Aspects of Addition of Angular Momentum
		20.8. Hydrogen Atom and Hidden Symmetry
		Problems
	21. Identical Particles
		21.1. Identical Particles and Exchange Degeneracy
		21.2. Permutation Operators
		21.3. Complete Symmetrizer and Antisymmetrizer
		21.4. The Symmetrization Postulate
		21.5. Building Symmetrized States and Probabilities
		21.6. Particles with Two Sets of Degrees of Freedom
		21.7. States of Two-Electron Systems
		21.8. Occupation Numbers
		Problems
III: Applications
	22. Density Matrix and Decoherence
		22.1. Ensembles and Mixed States
		22.2. The Density Matrix
		22.3. Dynamics of Density Matrices
		22.4. Subsystems and Schmidt Decomposition
		22.5. Open Systems and Decoherence
		22.6. The Lindblad Equation
		22.7. A Theory of Measurement?
		Problems
	23. Quantum Computation
		23.1. Qubits and Gates
		23.2. Deutsch’s Computation
		23.3. Grover’s Algorithm
		Problems
	24. Charged Particles in Electromagnetic Fields
		24.1. Electromagnetic Potentials
		24.2. Schrödinger Equation with Electromagnetic Potentials
		24.3. Heisenberg Picture
		24.4. Magnetic Fields on a Torus
		24.5. Particles in Uniform Magnetic Field: Landau Levels
		24.6. The Pauli Equation
		24.7. The Dirac Equation
		Problems
	25. Time-Independent Perturbation Theory
		25.1. Time-Independent Perturbations
		25.2. Nondegenerate Perturbation Theory
		25.3. The Anharmonic Oscillator
		25.4. Degenerate Perturbation Theory
		25.5. Degeneracy Lifted at Second Order
		25.6. Review of Hydrogen Atom
		25.7. Fine Structure of Hydrogen
		25.8. Zeeman Effect
		Problems
	26. WKB and Semiclassical Approximation
		26.1. The Classical Limit
		26.2. WKB Approximation Scheme
		26.3. Using Connection Formulae
		26.4. Airy Functions and Their Expansions
		26.5. Connection Formulae Derived
		26.6. Tunneling through a Barrier
		26.7. Double-Well Potentials
		Problems
	27. Time-Dependent Perturbation Theory
		27.1. Time-Dependent Hamiltonians
		27.2. The Interaction Picture
		27.3. Perturbative Solution in the Interaction Picture
		27.4. Constant Perturbations
		27.5. Harmonic Perturbations
		27.6. Fermi’s Golden Rule
		27.7. Helium Atom and Autoionization
		27.8. Modeling the Decay of a Discrete State to the Continuum
		27.9. Ionization of Hydrogen
		27.10. Atoms and Light
		27.11. Atom-Light Dipole Interaction
		27.12. Selection Rules
		Problems
	28. Adiabatic Approximation
		28.1. Adiabatic Changes and Adiabatic Invariants
		28.2. From Classical to Quantum Adiabatic Invariants
		28.3. Instantaneous Energy Eigenstates
		28.4. Quantum Adiabatic Theorem
		28.5. Landau-Zener Transitions
		28.6. Berry’s Phase
		28.7. Born-Oppenheimer Approximation
		28.8. The Hydrogen Molecule Ion
		Problems
	29. Scattering in One Dimension
		29.1. Scattering on the Half Line
		29.2. Time Delay
		29.3. Levinson’s Theorem
		29.4. Resonances
		29.5. Modeling Resonances
		Problems
	30. Scattering in Three Dimensions
		30.1. Energy Eigenstates for Scattering
		30.2. Cross Sections from Scattering Amplitudes
		30.3. Scattering Amplitude in Terms of Phase Shifts
		30.4. Computation of Phase Shifts
		30.5. Integral Equation for Scattering
		30.6. The Born Approximation
		Problems
References
Index
Useful Formulae