The Structure of Quantum Mechanics

Chapters

1. The wave function is not the state
   • One state in three representations
   • Amplitudes add before they square
   • The Born rule in general form
   • Global phase and relative phase
   • Superposition against mixture
   • The Schrodinger equation as a component statement
   • What is now settled about the wave function
   • Summary
   • Problems
2. Hilbert space and Dirac notation
   • Inner products and the space of states
   • Kets, bras and the antilinear correspondence
   • Orthonormal basis, components and Parseval
   • Completeness and the resolution of the identity
   • Change of basis as change of description
   • What infinite dimension costs
   • Summary
   • Problems
3. Observables, spectra and the spectral theorem
   • Linear operators, matrices and adjoints
   • Self-adjoint against Hermitian
   • The spectral theorem and what it is for
   • A degenerate eigenspace and its projector
   • Continuum normalisation and improper vectors
   • Observables defined by the spectral theorem
   • Summary
   • Problems
4. The postulates, measurement and collapse
   • Three magnets and states as unit rays
   • Five postulates and the Born rule with projectors
   • State reduction and the experiment that forces it
   • Non-selective measurement, where kets run out
   • The measurement problem, stated exactly
   • Interpretive options and their prices
   • Summary
   • Problems
5. Compatible observables and uncertainty as a theorem
   • Three labels for hydrogen
   • Compatible observables and simultaneous diagonalisation
   • The complete set of commuting observables
   • The uncertainty relation with its anticommutator term
   • Uncertainty is not disturbance
   • The energy-time relation without a time operator
   • What is now settled about compatibility and uncertainty
   • Summary
   • Problems
6. Position, momentum and the origin of the canonical commutator
   • Which operator is momentum?
   • The translation operator and its generator
   • Momentum defined as the generator of translations
   • Canonical quantisation and its ambiguity
   • The bracket correspondence and its limits
   • Translation invariance and conservation
   • Summary
   • Problems
7. Time evolution as a unitary group, and the three pictures
   • The evolution operator and its group property
   • The propagator
   • The time-ordered exponential
   • The Heisenberg picture
   • The interaction picture
   • Why no experiment distinguishes the pictures
   • Summary
   • Problems
8. The oscillator as an algebra
   • The ladder algebra as a definition
   • The spectrum from the algebra alone
   • Matrix elements without integrals
   • The wave functions, as an afterthought
   • What has actually been solved
   • Summary
   • Problems
9. Coherent states, Ehrenfest's theorem and the classical limit
   • Ehrenfest's theorem and its correction
   • The coherent state and the displacement operator
   • Poissonian number statistics
   • Rigid motion, non-orthogonality and overcompleteness
   • When a system has a trajectory
   • Summary
   • Problems
10. The path integral
    • The propagator's phase as an action
    • Time slicing the propagator
    • The sum over paths
    • The free and quadratic path integrals
    • The classical limit as stationary phase
    • The phase a vector potential adds
    • Summary
    • Problems
11. Slow change, sudden change and the Berry phase
    • Instantaneous eigenstates and the adiabatic theorem
    • The validity condition and where it fails
    • The sudden approximation and tritium decay
    • Dynamical and geometric phase
    • Spin one-half in a slowly rotating field
    • Where the geometric phase turns up
    • Summary
    • Problems
12. Symmetries, generators and conservation laws
    • What a symmetry is
    • Generator, symmetry and conservation
    • Symmetry forces degeneracy
    • Parity
    • Lattice translation and Bloch's theorem
    • Time reversal and antiunitarity
    • Summary
    • Problems
13. Angular momentum from the rotation group
    • What was thrown away
    • The generators of rotation and their algebra
    • The orbital case and the integer condition
    • Spherical harmonics as representation matrix elements
    • Finite rotations and the Wigner d matrices
    • Summary
    • Problems
14. Spin one-half, SU(2) and two-level dynamics
    • SU(2) as the double cover of SO(3)
    • The Pauli algebra from the group
    • The Bloch sphere
    • The rotating frame and Rabi oscillation
    • Observing the 2π sign
    • Summary
    • Problems
15. Adding angular momenta, tensor operators and Wigner-Eckart
    • The tensor product of two systems
    • The coupled basis and the decomposition rule
    • Building the Clebsch-Gordan coefficients
    • Irreducible tensor operators
    • The Wigner-Eckart theorem
    • What Part III bought
    • Summary
    • Problems
16. Central potentials as a rotation problem
    • Separation as rotational decomposition
    • The radial equation, with the ordering settled
    • The node theorem and radial ordering
    • Spherical wells and the isotropic oscillator's extra degeneracy
    • What is now settled about central potentials
    • Summary
    • Problems
17. Hydrogen, and the symmetry that explains it
    • The radial decomposition and hydrogen's nine states
    • The classical Runge-Lenz vector
    • The quantum Runge-Lenz vector and the SO(4) algebra
    • The spectrum with no differential equation
    • Which perturbation breaks which symmetry
    • Where the non-relativistic Coulomb problem stops
    • Summary
    • Problems
18. Charged particles in electromagnetic fields
    • Minimal coupling of a charge to a field
    • Gauge covariance of the theory and kinetic momenta
    • Landau levels from the algebra
    • The Aharonov-Bohm effect
    • What the vector potential turns out to be
    • The end of the argument
    • Summary
    • Problems