Chapter
1.6.8 Coherent and incoherent neutron scattering by a crystal
2 The mathematics of quantum mechanics I: finite dimension
2.1 Hilbert spaces of finite dimension
2.2 Linear operators on H
2.2.1 Linear, Hermitian, unitary operators
2.2.2 Projection operators and Dirac notation
2.3 Spectral decomposition of Hermitian operators
2.3.1 Diagonalization of a Hermitian operator
2.3.2 Diagonalization of a 2×2 Hermitian matrix
2.3.3 Complete sets of compatible operators
2.3.4 Unitary operators and Hermitian operators
2.3.5 Operator-valued functions
2.4.1 The scalar product and the norm
2.4.2 Commutators and traces
2.4.3 The determinant and the trace
2.4.5 The projection theorem
2.4.6 Properties of projectors
2.4.7 The Gaussian integral
2.4.8 Commutators and a degenerate eigenvalue
2.4.11 Operator identities
3 Polarization: photons and spin-1/2 particles
3.1 The polarization of light and photon polarization
3.1.1 The polarization of an electromagnetic wave
3.1.2 The photon polarization
3.1.3 Quantum cryptography
3.2.1 Angular momentum and magnetic moment in classical physics
3.2.2 The Stern–Gerlach experiment and Stern–Gerlach filters
3.2.3 Spin states of arbitrary orientation
3.2.4 Rotation of spin 1/2
3.2.5 Dynamics and time evolution
3.3.1 Decomposition and recombination of polarizations
3.3.2 Elliptical polarization
3.3.3 Rotation operator for the photon spin
3.3.4 Other solutions of (3.45)
3.3.5 Decomposition of a 2×2 matrix
3.3.6 Exponentials of Pauli matrices and rotation operators
3.3.7 The tensor Epsilonijk
3.3.8 A 2 rotation of spin 1/2
3.3.9 Neutron scattering by a crystal: spin-1/2 nuclei
4 Postulates of quantum physics
4.1 State vectors and physical properties
4.1.1 The superposition principle
4.1.2 Physical properties and measurement
4.1.3 Heisenberg inequalities II
4.2.1 The evolution equation
4.2.2 The evolution operator
4.2.4 The temporal Heisenberg inequality
4.2.5 The Schrödinger and Heisenberg pictures
4.3 Approximations and modeling
4.4.1 Dispersion and eigenvectors
4.4.2 The variational method
4.4.3 The Feynman–Hellmann theorem
4.4.4 Time evolution of a two-level system
4.4.6 The solar neutrino puzzle
4.4.7 The Schrödinger and Heisenberg pictures
4.4.8 The system of neutral K mesons
5 Systems with a finite number of levels
5.1 Elementary quantum chemistry
5.1.1 The ethylene molecule
5.1.2 The benzene molecule
5.2 Nuclear magnetic resonance (NMR)
5.2.1 A spin 1/2 in a periodic magnetic field
5.2.3 Principles of NMR and MRI
5.3.1 The ammonia molecule as a two-level system
5.3.2 The molecule in an electric field: the ammonia maser
5.3.3 Off-resonance transitions
5.5.1 An orthonormal basis of eigenvectors
5.5.2 The electric dipole moment of formaldehyde
5.5.4 Eigenvectors of the Hamiltonian (5.47)
5.5.5 The hydrogen molecular ion H+2
5.5.6 The rotating-wave approximation in NMR
6.1 The tensor product of two vector spaces
6.1.1 Definition and properties of the tensor product
6.1.2 A system of two spins 1/2
6.2 The state operator (or density operator)
6.2.1 Definition and properties
6.2.2 The state operator for a two-level system
6.2.3 The reduced state operator
6.2.4 Time dependence of the state operator
6.2.5 General form of the postulates
6.3.3 Interference and entangled states
6.3.4 Three-particle entangled states (GHZ states)
6.4.1 Measurement and decoherence
6.4.2 Quantum information
6.5.1 Independence of the tensor product from the choice of basis
6.5.2 The tensor product of two 2×2 matrices
6.5.3 Properties of state operators
6.5.4 Fine structure and the Zeeman effect in positronium
6.5.5 Spin waves and magnons
6.5.6 Spin echo and level splitting in NMR
6.5.7 Calculation of E(a, b)
6.5.8 Bell inequalities involving photons
6.5.9 Two-photon interference
6.5.10 Interference of emission times
6.5.11 The Deutsch algorithm
7 Mathematics of quantum mechanics II: infinite dimension
7.1.2 Realizations of separable spaces of infinite dimension
7.2 Linear operators on H
7.2.1 The domain and norm of an operator
7.2.2 Hermitian conjugation
7.3 Spectral decomposition
7.3.1 Hermitian operators
7.4.1 Spaces of infinite dimension
7.4.2 Spectrum of a Hermitian operator
7.4.3 Canonical commutation relations
7.4.4 Dilatation operators and the conformal transformation
8 Symmetries in quantum physics
8.1 Transformation of a state in a symmetry operation
8.1.1 Invariance of probabilities in a symmetry operation
8.2 Infinitesimal generators
8.2.3 Commutation relations of infinitesimal generators
8.3 Canonical commutation relations
8.3.2 Explicit realization and von Neumann’s theorem
8.3.3 The parity operator
8.4.1 The Hamiltonian in dimension d = 1
8.4.2 The Hamiltonian in dimension d = 3
8.5.2 Rotations and SU(2)
8.5.3 Commutation relations between momentum and angular momentum
8.5.4 The Lie algebra of a continuous group
8.5.5 The Thomas–Reiche–Kuhn sum rule
8.5.6 The center of mass and the reduced mass
8.5.7 The Galilean transformation
9.1 Diagonalization of X and P and wave functions
9.1.1 Diagonalization of X
9.1.4 Evolution of a free wave packet
9.2 The Schrödinger equation
9.2.1 The Hamiltonian of the Schrödinger equation
9.2.2 The probability density and the probability current density
9.3 Solution of the time-independent Schrödinger equation
9.3.2 Reflection and transmission by a potential step
The potential step: total reflection
The potential step: reflection and transmission
9.3.3 The bound states of the square well
9.4.1 The transmission matrix
9.5 The periodic potential
9.6 Wave mechanics in dimension d = 3
9.6.2 The phase space and level density
9.6.3 The Fermi Golden Rule
9.7.1 The Heisenberg inequalities
9.7.2 Wave-packet spreading
9.7.3 A Gaussian wave packet
9.7.4 Heuristic estimates using the Heisenberg inequality
9.7.5 The Lennard–Jones potential for helium
9.7.7 A delta-function potential
9.7.8 Transmission by a well
9.7.9 Energy levels of an infinite cubic well in dimension d = 3
9.7.10 The probability current in three dimensions
9.7.12 The Fermi Golden Rule
9.7.13 Study of the Stern–Gerlach experiment
9.7.14 The von Neumann model of measurement
9.7.15 The Galilean transformation
10.1 Diagonalization of J2 and Jz
10.3 Orbital angular momentum
10.3.1 The orbital angular momentum operator
10.3.2 Properties of the spherical harmonics
1. Basis on the unit sphere
2. Relation to the Legendre polynomials
3. Transformation under rotation
4. Parity of the spherical harmonics
10.4 Particle in a central potential
10.4.1 The radial wave equation
10.5 Angular distributions in decays
10.5.1 Rotations by pi, parity, and reflection with respect to a plane
10.5.2 Dipole transitions
10.5.3 Two-body decays: the general case
10.6 Addition of two angular momenta
10.6.1 Addition of two spins 1/2
10.6.2 The general case: addition of two angular momenta J1 and J2
10.6.3 Composition of rotation matrices
10.6.4 The Wigner–Eckart theorem (scalar and vector operators)
10.7.2 Rotation of angular momentum
10.7.3 Rotations (theta, phi)
10.7.4 The angular momenta…
10.7.5 Orbital angular momentum
10.7.6 Relation between the rotation matrices and the spherical harmonics
10.7.7 Independence of the energy from m
10.7.8 The spherical well
10.7.9 The hydrogen atom for…
10.7.10 Matrix elements of a potential
10.7.11 The radial equation in dimension d = 2
10.7.12 Symmetry property of the matrices d(j)
10.7.14 Measurement of the Lambda0 magnetic moment
10.7.15 Production and decay of the rho+ meson
10.7.16 Interaction of two dipoles
10.7.18 Irreducible tensor operators
11 The harmonic oscillator
11.1 The simple harmonic oscillator
11.1.1 Creation and annihilation operators
11.1.2 Diagonalization of the Hamiltonian
11.1.3 Wave functions of the harmonic oscillator
11.3 Introduction to quantized fields
11.3.1 Sound waves and phonons
11.3.2 Quantization of a scalar field in one dimension
11.3.3 Quantization of the electromagnetic field
11.3.4 Quantum fluctuations of the electromagnetic field
11.4 Motion in a magnetic field
11.4.1 Local gauge invariance
11.4.2 A uniform magnetic field: Landau levels
11.5.1 Matrix elements of Q and P
11.5.2 Mathematical properties
11.5.4 Coupling to a classical force
11.5.6 Zero-point energy of the Debye model
11.5.7 The scalar and vector potentials in Coulomb gauge
11.5.8 Commutation relations and Hamiltonian of the electromagnetic field
11.5.9 Quantization in a cavity
11.5.10 Current conservation in the presence of a magnetic field
11.5.11 Non-Abelian gauge transformations
11.5.12 The Casimir effect
11.5.13 Quantum computing with trapped ions
12 Elementary scattering theory
12.1 The cross section and scattering amplitude
12.1.1 The differential and total cross sections
12.1.2 The scattering amplitude
12.2 Partial waves and phase shifts
12.2.1 The partial-wave expansion
12.2.2 Low-energy scattering
12.2.3 The effective potential
12.2.4 Low-energy neutron–proton scattering
12.3 Inelastic scattering
12.3.1 The optical theorem
12.3.2 The optical potential
12.4.1 The integral equation of scattering
12.4.2 Scattering of a wave packet
12.5.2 Low-energy neutron scattering by a hydrogen molecule
12.5.3 Analytic properties of the neutron–proton scattering amplitude
12.5.4 The Born approximation
12.5.6 The cross section for neutrino absorption
13.1.1 Symmetry or antisymmetry of the state vector
13.1.2 Spin and statistics
13.2 The scattering of identical particles
13.4.1 The Tonos- particle and color
13.4.2 Parity of the pi meson
13.4.3 Spin-1/2 fermions in an infinite well
13.4.5 Quantum statistics and beam splitters
14.1 Approximation methods
14.1.2 Nondegenerate perturbation theory
14.1.3 Degenerate perturbation theory
14.1.4 The variational method
14.2.1 Energy levels in the absence of spin
14.2.2 The fine structure
14.2.4 The hyperfine structure
14.3 Atomic interactions with an electromagnetic field
14.3.1 The semiclassical theory
14.3.2 The dipole approximation
14.3.3 The photoelectric effect
14.3.4 The quantized electromagnetic field: spontaneous emission
14.4 Laser cooling and trapping of atoms
14.4.1 The optical Bloch equations
14.4.2 Dissipative forces and reactive forces
14.4.4 A magneto-optical trap
14.5 The two-electron atom
14.5.1 The ground state of the helium atom
14.5.2 The excited states of the helium atom
14.6.1 Second-order perturbation theory and van der Waals forces
14.6.2 Order-alpha2 corrections to the energy levels
14.6.5 The diamagnetic term
14.6.6 Vacuum Rabi oscillations
14.6.8 Radiative capture of neutrons by hydrogen
15.1 Generalized measurements
15.1.1 Schmidt’s decomposition
15.1.2 Positive operator-valued measures
15.1.3 Example: a POVM with spins 1/2
15.2.1 Kraus decomposition
15.2.2 The depolarizing channel
15.2.3 The phase-damping channel
15.2.4 The amplitude-damping channel
15.3 Master equations: the Lindblad form
15.3.1 The Markovian approximation
15.3.2 The Lindblad equation
15.3.3 Example: the damped harmonic oscillator
15.4 Coupling to a thermal bath of oscillators
15.4.1 Exact evolution equations
15.4.2 The Markovian approximation
15.4.3 Relaxation of a two-level system
15.4.4 Quantum Brownian motion
15.4.5 Decoherence and Schrödinger’s cats
15.5.1 POVM as projective measurement in a direct sum
15.5.2 Using a POVM to distinguish between states
15.5.3 A POVM on two arbitrary qubit states
15.5.4 Transposition is not completely positive
15.5.5 Phase and amplitude damping
15.5.6 Details of the proof of the master equation
15.5.7 Superposition of coherent states
15.5.8 Dissipation in a two-level system
15.5.9 Simple models of relaxation
15.5.10 Another choice for the spectral function J(omega)
15.5.11 The Fokker–Planck–Kramers equation for a Brownian particle
Appendix A The Wigner theorem and time reversal
Appendix B Measurement and decoherence
B.1 An elementary model of measurement
B.3 Interaction with a field inside the cavity
Appendix C The Wigner–Weisskopf method
Quantum Physics
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