Quantum Mechanics

Cover
Half-title
Endorsement
Title page
Copyright information
Contents
Preface
Book Organisation
Acknowledgements
Part I Basics
	1 Stories and Thoughts about Quantum Mechanics
		Further Reading
	2 Quantum States
		2.1 States of a Quantum System
			2.1.1 Kets
			2.1.2 Scalar Product
			2.1.3 Bras
			2.1.4 Norm
			2.1.5 Orthogonality
			2.1.6 Operators
			2.1.7 Bases
			2.1.8 Tensor Product
		2.2 Two-State Systems
		2.3 The Wave Function of a One-Dimensional System
			2.3.1 Discretised System
			2.3.2 Continuum System
		Summary
		Problems
	3 Observables
		3.1 Introducing Observables
			3.1.1 A First Example: Position Operator
			3.1.2 Generic Observables
		3.2 Observing Observables
		3.3 Hermitian Operators
			3.3.1 Hermitian Conjugate
			3.3.2 Hermitian Operators
		3.4 Properties of Hermitian Operators
		3.5 Spectral Decomposition
		3.6 Collapse of the State Vector
		3.7 Compatible Observables
		3.8 Complete Sets of Commuting Observables
		3.9 Continuous Spectrum
			3.9.1 Eigenvalue Equation
			3.9.2 Orthonormality
			3.9.3 Spectral Decomposition
		3.10 Momentum Operator
			3.10.1 Momentum as Generator of Translations
			3.10.2 Position and Momentum Representations
			3.10.3 Wave Packets
		3.11 The Uncertainty Principle
		Summary
		Problems
	4 Dynamics
		4.1 Schrödinger Equation
			4.1.1 Equation of Motion
		4.2 Eigenstates of the Hamiltonian
		4.3 Evolution of a Generic State
		4.4 One-Dimensional System
		4.5 Some Properties of One-Dimensional Potentials
			4.5.1 General Restrictions on Ψ(x,t)
			4.5.2 Energy Expectation Value
			4.5.3 Ehrenfest’s Theorem
			4.5.4 Degeneracy
			4.5.5 Nodes
		4.6 Probability Current
		Summary
		Problems
	5 Potentials
		5.1 Potential Step
		5.2 Tunnelling
		5.3 Infinite Potential Well
		5.4 Symmetry under Parity
		5.5 Finite Potential Well
		Summary
		Problems
	6 Harmonic Oscillator
		6.1 The Harmonic Oscillator in Classical Mechanics
		6.2 The Quantum Harmonic Oscillator
		6.3 Factorising the Hamiltonian
		6.4 Creation and Annihilation
		6.5 Eigensystem
			6.5.1 Eigenvalues
			6.5.2 Normalisation of Eigenstates
			6.5.3 Wave Functions
		6.6 Brute Force Solution
		Summary
		Problems
	7 Systems in Three Spatial Dimensions
		7.1 Quantum States
		7.2 Observables
			7.2.1 General Considerations
			7.2.2 Position
			7.2.3 Momentum
		7.3 Dynamics
		7.4 Separation of Variables: Three-Dimensional Harmonic Oscillator
		7.5 Degeneracy
		Summary
		Problems
	8 Angular Momentum
		8.1 Angular Momentum Operator
		8.2 Squared Norm of the Angular Momentum
		8.3 Eigenfunctions
		8.4 Physical Interpretation
		8.5 Algebraic Solution of the Eigenvalue Equations
		8.6 Nomenclature
		8.7 Normalisation
		8.8 Matrix Representations
		8.9 Wave Functions
		Summary
		Problems
	9 Spin
		9.1 The Stern–Gerlach Experiment
		9.2 Physical States
		9.3 Matrix Representation
		9.4 Eigenvectors
		9.5 Scalar Products
		9.6 Eigenvectors of [sub(x)]
		9.7 The Stern–Gerlach Experiment Reloaded
		Summary
		Problems
	10 Addition of Angular Momenta
		10.1 Total Angular Momentum Operator
		10.2 Addition Theorem
		10.3 Example: Two Spin-1/2 Particles
		Summary
		Problems
	11 Central Potentials
		11.1 Stationary States
		11.2 Physical Interpretation
		11.3 Quantum Rotator
		11.4 Central Square Well
		Summary
		Problems
	12 Hydrogen Atom
		12.1 Stationary States
		12.2 Solution of the Radial Equation
		12.3 Physical Interpretation
		Summary
		Problems
	13 Identical Particles
		13.1 Permutation Symmetry
		13.2 A First Look at Helium
		13.3 Two-Electron Wave Function
		13.4 More on the Helium Atom
		13.5 Pauli Exclusion Principle
		Summary
		Problems
	14 Symmetries in Quantum Mechanics
		14.1 Classical Symmetry
		14.2 Quantum Symmetry
		14.3 Symmetry Groups in Physics
		14.4 Group of Translations
		14.5 Group of Rotations
			14.5.1 Rotations in Two Dimensions
			14.5.2 Rotations in Three Dimensions
		14.6 Rotations in the Space of Quantum States
		14.7 Rotations Acting on Operators
		14.8 Commutation Relations for a Generic Non-Abelian Group
		Summary
		Problems
Part II Applications
	15 Quantum Entanglement
		15.1 Hidden Variables and the Einstein–Podolsky–Rosen Paradox
			15.1.1 The Einstein–Podolsky–Rosen Paradox
		15.2 Bell’s Inequality
			15.2.1 Spin Correlations
			15.2.2 Experimental Tests of Bell’s Inequality
		15.3 Characterising Entanglement
			15.3.1 Bit and Qubit
			15.3.2 Information Theory Essentials – Shannon Entropy
			15.3.3 Von Neumann Entropy
			15.3.4 Entanglement Entropy
			15.3.5 Other Measures of Information
		15.4 Quantum Communication
			15.4.1 No-Cloning Theorem
			15.4.2 Quantum Teleportation of a Qubit
			15.4.3 Superdense Coding
		15.5 Quantum Computing
			15.5.1 Quantum Register and Logic Gates
			15.5.2 Deutsch’s Algorithm
			15.5.3 Grover’s Algorithm
		15.A Appendix: Klein’s Inequality
		Summary
		Reference
		Further Reading
		Problems
	16 Time-Independent Perturbation Theory
		16.1 Nondegenerate Time-Independent Perturbation Theory
			16.1.1 First-Order Correction
			16.1.2 Second-Order Energy Correction
			16.1.3 Normalisation and Orthogonality of the Eigenstates to First Order
			16.1.4 Higher-Order Corrections
			16.1.5 Properties of the Perturbation Expansion
		16.2 Degenerate Perturbation Theory
			16.2.1 First-Order Degenerate Perturbation Correction
			16.2.2 Second-Order Degenerate Perturbation Energy Correction
		16.3 Applications
			16.3.1 Hydrogen Fine Structure
			16.3.2 Helium Atom Perturbation Treatment of Electron Repulsion Term
		16.A Appendix: Derivation of the Fine-Structure Terms from Relativistic Theory
		Summary
		References
		Further Reading
		Problems
	17 Calculation Methods Beyond Perturbation Theory
		17.1 Rayleigh–Ritz Variational Method
			17.1.1 Ground State of Hydrogen
			17.1.2 Ground State of Helium
			17.1.3 Excited States
		17.2 Atomic Systems
			17.2.1 First Excited States of Helium
			17.2.2 Multi-electron Atoms
		17.3 Born–Oppenheimer Approximation
			17.3.1 The H[sub(2)sup(+)] Ion and Bonding
		17.4 Hellmann–Feynman Method
		17.5 Wenzel–Kramers–Brillouin–Jeffreys Approximation
			17.5.1 Potential Barrier
			17.5.2 Potential Well
			17.5.3 Symmetric Double Well
		Summary
		Further Reading
		Problems
	18 Time-Dependent Perturbation Theory
		18.1 Time-Dependent Hamiltonians
			18.1.1 Time-Dependent Perturbation Theory
		18.2 Time-Dependent Perturbations
			18.2.1 Time-Independent Perturbation Switched on at a Given Time
			18.2.2 Fermi’s Golden Rule
			18.2.3 Harmonic Perturbations
		18.3 Interaction of Radiation with Quantum Systems
			18.3.1 Semiclassical Treatment of Electromagnetic Radiation
			18.3.2 Interaction with a One-Electron Atom
			18.3.3 Selection Rules
		18.4 Time-Dependent Perturbation Theory at Higher Order
			18.4.1 Time Dependence in Quantum Mechanics
			18.4.2 Time Evolution Operator in the Interaction Picture
		Summary
		Problems
	19 Quantum Scattering Theory
		19.1 Scattering Kinematics
			19.1.1 The Differential Cross-Section
		19.2 Evolution of a Wave Packet
		19.3 Time-Dependent Approach to Scattering
			19.3.1 The Born Approximation
			19.3.2 Two-Body Scattering
		19.4 Time-Independent Approach to Scattering
			19.4.1 Relation of Wave Packet to Time-Independent Approach
			19.4.2 The Scattering Amplitude
			19.4.3 Green Function Method for Calculating  f(θ, [phi])
			19.4.4 Partial Wave Analysis
		19.A Appendix: Solutions of the Radial Equation for a Free Particle
		Summary
		Further Reading
		Problems
Appendix: Gaussian/SI Electromagnetic Units
Index