Quantum Mechanics: A New Introduction

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Half-title
Title
Copyright
Preface
Contents
I Basic quantum mechanics
	1 Introduction
		1.1 The quantum behavior of the electron
			1.1.1 Diffraction and interference—visualizing the quantum world
			1.1.2 The stability and identity of atoms
			1.1.3 Tunnel effects
		1.2 The birth of quantum mechanics
			1.2.1 From the theory of specific heat to Planck’s formula
			1.2.2 The photoelectric effect
			1.2.3 Bohr’s atomic model
			1.2.4 The Bohr–Sommerfeld quantization condition; de Broglie’s wave
	Further reading
	Guide to the Supplements
	Problems
	Numerical analyses
	2 Quantum mechanical laws
		2.1 Quantum states
			2.1.1 Composite systems
			2.1.2 Photon polarization and the statistical nature of quantum mechanics
		2.2 The uncertainty principle
		2.3 The fundamental postulate
			2.3.1 The projection operator and state vector reduction
			2.3.2 Hermitian operators
			2.3.3 Products of operators, commutators, and compatible observables
			2.3.4 The position operator, the momentum operator, fundamental commutators, and Heisenberg’s relation
			2.3.5 Heisenberg’s relations
		2.4 The Schrödinger equation
			2.4.1 More about the Schrödinger equations
			2.4.2 The Heisenberg picture
		2.5 The continuous spectrum
			2.5.1 The delta function
			2.5.2 Orthogonality
			2.5.3 The position and momentum eigenstates; momentum as a translation operator
		2.6 Completeness
	Problems
	Numerical analyses
	3 The Schrödinger equation
		3.1 General properties
			3.1.1 Boundary conditions
			3.1.2 Ehrenfest’s theorem
			3.1.3 Current density and conservation of probability
			3.1.4 The virial and Feynman–Hellman theorems
		3.2 One-dimensional systems
			3.2.1 The free particle
			3.2.2 Topologically nontrivial space
			3.2.3 Special properties of one-dimensional Schrödinger equations
		3.3 Potential wells
			3.3.1 Infinitely deep wells (walls)
			3.3.2 The finite square well
			3.3.3 An application
		3.4 The harmonic oscillator
			3.4.1 The wave function and Hermite polynomials
			3.4.2 Creation and annihilation operators
		3.5 Scattering problems and the tunnel effect
			3.5.1 The potential barrier and the tunnel effect
			3.5.2 The delta function potential
			3.5.3 General aspects of the scattering problem
		3.6 Periodic potentials
			3.6.1 The band structure of the energy spectrum
			3.6.2 Analysis
	Guide to the Supplements
	Problems
	Numerical analyses
	4 Angular momentum
		4.1 Commutation relations
		4.2 Space rotations
		4.3 Quantization
		4.4 The Stern–Gerlach experiment
		4.5 Spherical harmonics
		4.6 Matrix elements of J
			4.6.1 Spin- ½  and Pauli matrices
		4.7 The composition rule
			4.7.1 The Clebsch–Gordan coefficients
		4.8 Spin
			4.8.1 Rotation matrices for spin ½
	Guide to the Supplements
	Problems
	5 Symmetry and statistics
		5.1 Symmetries in Nature
		5.2 Symmetries in quantum mechanics
			5.2.1 The ground state and symmetry
			5.2.2 Parity (P)
			5.2.3 Time reversal
			5.2.4 The Galilean transformation
			5.2.5 The Wigner–Eckart theorem
		5.3 Identical particles: Bose–Einstein and Fermi–Dirac statistics
			5.3.1 Identical bosons
			5.3.2 Identical fermions and Pauli’s exclusion principle
	Guide to the Supplements
	Problems
	6 Three-dimensional problems
		6.1 Simple three-dimensional systems
			6.1.1 Reduced mass
			6.1.2 Motion in a spherically symmetric potential
			6.1.3 Spherical waves
		6.2 Bound states in potential wells
		6.3 The three-dimensional oscillator
		6.4 The hydrogen atom
	Guide to the Supplements
	Problems
	Numerical analyses
	7 Some finer points of quantum mechanics
		7.1 Representations
			7.1.1 Coordinate and momentum representations
		7.2 States and operators
			7.2.1 Bra and ket; abstract Hilbert space
		7.3 Unbounded operators
			7.3.1 Self-adjoint operators
		7.4 Unitary transformations
		7.5 The Heisenberg picture
			7.5.1 The harmonic oscillator in the Heisenberg picture
		7.6 The uncertainty principle
		7.7 Mixed states and the density matrix
			7.7.1 Photon polarization
		7.8 Quantization in general coordinates
	Further reading
	Guide to the Supplements
	Problems
	8 Path integrals
		8.1 Green functions
		8.2 Path integrals
			8.2.1 Derivation
			8.2.2 Mode expansion
			8.2.3 Feynman graphs
			8.2.4 Back to ordinary (Minkowski) time
			8.2.5 Tunnel effects and instantons
	Further reading
	Numerical analyses
II Approximation methods
	9 Perturbation theory
		9.1 Time-independent perturbations
			9.1.1 Degenerate levels
			9.1.2 The Stark effect on the n = 2 level of the hydrogen atom
			9.1.3 Dipole interactions and polarizability
		9.2 Quantum transitions
			9.2.1 Perturbation lasting for a finite interval
			9.2.2 Periodic perturbation
			9.2.3 Transitions in a discrete spectrum
			9.2.4 Resonant oscillation between two levels
		9.3 Transitions in the continuum
			9.3.1 State density
		9.4 Decays
		9.5 Electromagnetic transitions
			9.5.1 The dipole approximation
			9.5.2 Absorption of radiation
			9.5.3 Induced (or stimulated) emission
			9.5.4 Spontaneous emission
		9.6 The Einstein coefficients
	Guide to the Supplements
	Problems
	Numerical analyses
	10 Variational methods
		10.1 The variational principle
			10.1.1 Lower limits
			10.1.2 Truncated Hilbert space
		10.2 Simple applications
			10.2.1 The harmonic oscillator
			10.2.2 Helium: an elementary variational calculation
			10.2.3 The virial theorem
		10.3 The ground state of the helium
	Guide to the Supplements
	Problems
	Numerical analyses
	11 The semi-classical approximation
		11.1 The WKB approximation
			11.1.1 Connection formulas
		11.2 The Bohr–Sommerfeld quantization condition
			11.2.1 Counting the quantum states
			11.2.2 Potentials defined for x > 0 only
			11.2.3 On the meaning of the limit _ ? 0
			11.2.4 Angular variables
			11.2.5 Radial equations
			11.2.6 Examples
		11.3 The tunnel effect
			11.3.1 The double well
			11.3.2 The semi-classical treatment of decay processes
			11.3.3 The Gamow–Siegert theory
		11.4 Phase shift
	Further reading
	Guide to the Supplements
	Problems
	Numerical analyses
III Applications
	12 Time evolution
		12.1 General features of time evolution
		12.2 Time-dependent unitary transformations
		12.3 Adiabatic processes
			12.3.1 The Landau–Zener transition
			12.3.2 The impulse approximation
			12.3.3 The Berry phase
			12.3.4 Examples
		12.4 Some nontrivial systems
			12.4.1 A particle within moving walls
			12.4.2 Resonant oscillations
			12.4.3 A particle encircling a solenoid
			12.4.4 A ring with a defect
		12.5 The cyclic harmonic oscillator: a theorem
			12.5.1 Inverse linear variation of the frequency
			12.5.2 The Planck distribution inside an oscillating cavity
			12.5.3 General power-dependent frequencies
			12.5.4 Exponential dependence
			12.5.5 Creation and annihilation operators; coupled oscillators
	Guide to the Supplements
	Problems
	Numerical analyses
	13 Metastable states
		13.1 Green functions
			13.1.1 Analytic properties of the resolvent
			13.1.2 Free particles
			13.1.3 The free Green function in general dimensions
			13.1.4 Expansion in powers of HI
		13.2 Metastable states
			13.2.1 Formulation of the problem
			13.2.2 The width of a metastable state; the mean halflifetime
			13.2.3 Formal treatment
		13.3 Examples
			13.3.1 Discrete–continuum coupling
		13.4 Complex scale transformations
			13.4.1 Analytic continuation
		13.5 Applications and examples
			13.5.1 Resonances in helium
			13.5.2 The potential Vor2e-r
			13.5.3 The unbounded potential; the Lo Surdo–Stark effect
	Further reading
	Problems
	Numerical analyses
	14 Electromagnetic interactions
		14.1 The charged particle in an electromagnetic field
			14.1.1 Classical particles
			14.1.2 Quantum particles in electromagnetic fields
			14.1.3 Dipole and quadrupole interactions
			14.1.4 Magnetic interactions
			14.1.5 Relativistic corrections: LS coupling
			14.1.6 Hyperfine interactions
		14.2 The Aharonov–Bohm effect
			14.2.1 Superconductors
		14.3 The Landau levels
			14.3.1 The quantum Hall effect
		14.4 Magnetic monopoles
	Guide to the Supplements
	Problems
	Numerical analyses
	15 Atoms
		15.1 Electronic configurations
			15.1.1 The ionization potential
			15.1.2 The spectrum of alkali metals
			15.1.3 X rays
		15.2 The Hartree approximation
			15.2.1 Self-consistent fields and the variational principle
			15.2.2 Some results
		15.3 Multiplets
			15.3.1 Structure of the multiplets
		15.4 Slater determinants
		15.5 The Hartree–Fock approximation
			15.5.1 Examples
		15.6 Spin–orbit interactions
			15.6.1 The hydrogen atom
		15.7 Atoms in external electric fields
			15.7.1 Dipole interaction and polarizability
			15.7.2 Quadrupole interactions
		15.8 The Zeeman effect
			15.8.1 The Zeeman effect in quantum mechanics
	Further reading
	Guide to the Supplements
	Problems
	Numerical analyses
	16 Elastic scattering theory
		16.1 The cross section
		16.2 Partial wave expansion
			16.2.1 The semi-classical limit
		16.3 The Lippman–Schwinger equation
		16.4 The Born approximation
		16.5 The eikonal approximation
		16.6 Low-energy scattering
		16.7 Coulomb scattering: Rutherford’s formula
			16.7.1 Scattering of identical particles
	Further reading
	Guide to the Supplements
	Problems
	Numerical analyses
	17 Atomic nuclei and elementary particles
		17.1 Atomic nuclei
			17.1.1 General features
			17.1.2 Isospin
			17.1.3 Nuclear forces, pion exchange, and the Yukawa potential
			17.1.4 Radioactivity
			17.1.5 The deuteron and two-nucleon forces
		17.2 Elementary particles: the need for relativistic quantum field theories
			17.2.1 The Klein–Gordon and Dirac equations
			17.2.2 Quantization of the free Klein–Gordon fields
			17.2.3 Quantization of the free Dirac fields and the spin– statistics connection
			17.2.4 Causality and locality
			17.2.5 Self-interacting scalar fields
			17.2.6 Non-Abelian gauge theories: the Standard Model
	Further reading
IV Entanglement and Measurement
	18 Quantum entanglement
		18.1 The EPRB Gedankenexperiment and quantum entanglement
		18.2 Aspect’s experiment
		18.3 Entanglement with more than two particles
		18.4 Factorization versus entanglement
		18.5 A measure of entanglement: entropy
	Further reading
	19 Probability and measurement
		19.1 The probabilistic nature of quantum mechanics
		19.2 Measurement and state preparation: from PVM to POVM 519
		19.3 Measurement “problems”
			19.3.1 The EPR “paradox”
			19.3.2 Measurement as a physical process: decoherence and the classical limit
			19.3.3 Schrödinger’s cat
			19.3.4 The fundamental postulate versus Schr¨odinger’s equation
			19.3.5 Is quantum mechanics exact?
			19.3.6 Cosmology and quantum mechanics
		19.4 Hidden-variable theories
			19.4.1 Bell’s inequalities
			19.4.2 The Kochen–Specker theorem
			19.4.3 “Quantumnon-locality” versus “locally causal theories” or “local realism”
	Further reading
	Guide to the Supplements
V Supplements
	20 Supplements for Part I
		20.1 Classical mechanics
			20.1.1 The Lagrangian formalism
			20.1.2 The Hamiltonian (canonical) formalism
			20.1.3 Poisson brackets
			20.1.4 Canonical transformations
			20.1.5 The Hamilton–Jacobi equation
			20.1.6 Adiabatic invariants
			20.1.7 The virial theorem
		20.2 The Hamiltonian of electromagnetic radiation field in the vacuum
		20.3 Orthogonality and completeness in a system with a onedimensional delta function potential
			20.3.1 Orthogonality
			20.3.2 Completeness
		20.4 The S matrix; the wave packet description of scattering
			20.4.1 The wave packet description
		20.5 Legendre polynomials
		20.6 Groups and representations
			20.6.1 Group axioms; some examples
			20.6.2 Group representations
			20.6.3 Lie groups and Lie algebras
			20.6.4 The U(N) group and the quarks
		20.7 Formulas for angular momentum
		20.8 Young tableaux
		20.9 N-particle matrix elements
		20.10 The Fock representation
			20.10.1 Bosons
			20.10.2 Fermions
		20.11 Second quantization
		20.12 Supersymmetry in quantum mechanics
		20.13 Two- and three-dimensional delta function potentials
			20.13.1 Bound states
			20.13.2 Self-adjoint extensions
			20.13.3 The two-dimensional delta-function potential: a quantum anomaly
		20.14 Superselection rules
		20.15 Quantum representations
			20.15.1 Weyl’s commutation relations
			20.15.2 Von Neumann’s theorem
			20.15.3 Angular variables
			20.15.4 Canonical transformations
			20.15.5 Self-adjoint extensions
		20.16 Gaussian integrals and Feynman graphs
	21 Supplements for Part II
		21.1 Supplements on perturbation theory
			21.1.1 Change of boundary conditions
			21.1.2 Two-level systems
			21.1.3 Van der Waals interactions
			21.1.4 The Dalgarno–Lewis method
		21.2 The fine structure of the hydrogen atom
			21.2.1 A semi-classical model for the Lamb shift
		21.3 Hydrogen hyperfine interactions
		21.4 Divergences of perturbative series
			21.4.1 Perturbative series at large orders: the anharmonic oscillator
			21.4.2 The origin of the divergence
			21.4.3 The analyticity domain
			21.4.4 Asymptotic series
			21.4.5 The dispersion relation
			21.4.6 The perturbative–variational approach
		21.5 The semi-classical approximation in general systems
			21.5.1 Introduction
			21.5.2 Keller quantization
			21.5.3 Integrable systems
			21.5.4 Examples
			21.5.5 Caustics
			21.5.6 The KAM theorem and quantization
	22 Supplements for Part III
		22.1 The K0–K0 system and CP violation
		22.2 Level density
			22.2.1 The free particle
			22.2.2 g(E) and the partition function
			22.2.3 g(E) and short-distance behavior
			22.2.4 Level density and scattering
			22.2.5 The stabilization method
		22.3 Thomas precession
		22.4 Relativistic corrections in an external field
		22.5 The Hamiltonian for interacting charged particles
			22.5.1 The interaction potentials
			22.5.2 Spin-dependent interactions
			22.5.3 The quantum Hamiltonian
			22.5.4 Electron–electron interactions
			22.5.5 Electron–nucleus interactions
			22.5.6 The 1/M corrections
		22.6 Quantization of electromagnetic fields
			22.6.1 Matrix elements
		22.7 Atoms
			22.7.1 The Thomas–Fermi approximation
			22.7.2 The Hartree approximation
			22.7.3 Slater determinants and matrix elements
			22.7.4 Hamiltonians for closed shells
			22.7.5 Mean energy
			22.7.6 Hamiltonians for incomplete shells
			22.7.7 Eigenvalues of H
			22.7.8 The elementary theory of multiplets
			22.7.9 The Hartree–Fock equations
			22.7.10The role of Lagrange multipliers
			22.7.11 Koopman’s theorem
		22.8 H2+
		22.9 The Gross–Pitaevski equation
		22.10 The semi-classical scattering amplitude
			22.10.1 Caustics and rainbows
	23 Supplements for Part IV
		23.1 Speakable and unspeakable in quantum mechanics
			23.1.1 Bell’s toy model for hidden variables
			23.1.2 Bohm’s pilot waves
			23.1.3 The many-worlds interpretation
			23.1.4 Spontaneous wave function collapse
	24 Mathematical appendices and tables
		24.1 Mathematical appendices
			24.1.1 Laplace’s method
			24.1.2 The saddle-point method
			24.1.3 Airy functions
References
Index