Quantum Mechanics

Quantum Mechanics in 28 chapters — from “Blackbody radiation and the photoelectric effect” to “The Dirac equation and quantum electrodynamics”.

Chapters

  1. Blackbody radiation and the photoelectric effect
    • Classical physics at the turn of the century
    • The ultraviolet catastrophe
    • Planck’s resolution
    • The photoelectric effect
    • Einstein 1905: light quanta
    • What was established, and what remained
    • Summary and forward view
    • Problems
  2. Atomic spectra and the Bohr model
    • Atomic spectra before Bohr
    • The Rutherford atom, 1911
    • Bohr’s postulates, 1913
    • Derivation of the Bohr spectrum
    • The Franck-Hertz experiment, 1914
    • What Bohr got right, and what he got wrong
    • The correspondence principle
    • Summary and forward view
    • Problems
  3. Wave-particle duality and matter waves
    • Light is a wave
    • Light is a particle, too
    • De Broglie’s hypothesis, 1924
    • Davisson and Germer, 1927
    • The double-slit experiment with electrons
    • Schrödinger’s wave mechanics
    • Born’s probability interpretation
    • Summary and forward view
    • Problems
  4. Spin and the Stern-Gerlach experiment
    • What angular momentum was in 1921
    • The Stern-Gerlach experiment, 1922
    • The anomalous Zeeman effect and doubled spectral lines
    • Pauli’s exclusion principle, 1925
    • Uhlenbeck and Goudsmit, 1925
    • The classical picture fails: Ehrenfest and Lorentz
    • Dirac 1928: spin from the relativistic wave equation
    • Summary and bridge to Part II
    • Problems
  5. Two-state systems and the Stern-Gerlach experiment
    • The Stern-Gerlach experiment as a datum
    • States as vectors
    • Observables as operators
    • Measurement and probability
    • The Pauli matrices
    • The Bloch sphere
    • Change of basis and the meaning of superposition
    • Summary and forward view
    • Problems
  6. Observables and measurement
    • From two states to many
    • Self-adjoint operators and the spectral theorem
    • Commuting observables and simultaneous eigenstates
    • Expectation values, variance, and uncertainty
    • Continuous spectra and infinite-dimensional Hilbert spaces
    • The Fourier transform as a basis change
    • Projective measurements and POVMs
    • Summary and forward view
    • Problems
  7. Composite systems and entanglement
    • Two systems, one Hilbert space
    • Product states versus entangled states
    • Partial trace and reduced density matrices
    • Schmidt decomposition
    • Measures of entanglement
    • Bell states, Bell measurements, and the Bell basis
    • The no-cloning theorem
    • Summary and forward view
    • Problems
  8. Time evolution and the Schrödinger equation
    • From kinematics to dynamics
    • The time-dependent Schrödinger equation
    • The time-evolution operator and unitarity
    • Energy eigenstates and the general solution
    • The Heisenberg picture
    • Three worked examples
    • Symmetries and conservation laws
    • Summary and forward view
    • Problems
  9. Density matrices and decoherence
    • Why we need density matrices
    • The density matrix formalism
    • Pure and mixed states
    • The Bloch ball revisited
    • Dynamics: the von Neumann equation
    • Open quantum systems and the Lindblad equation
    • Decoherence
    • Summary and forward view
    • Problems
  10. The harmonic oscillator
    • Why this chapter matters
    • The classical oscillator, briefly
    • The quantum oscillator: ladder operator solution
    • The ground state and wavefunctions
    • Heisenberg picture and classical correspondence
    • Coherent states
    • The oscillator everywhere: vibrations, phonons, photons
    • Summary and forward view
    • Problems
  11. Angular momentum
    • Why this chapter
    • The algebra of angular momentum
    • Eigenvalues from the algebra alone
    • Matrix elements and the first few representations
    • Orbital angular momentum and spherical harmonics
    • Rotations and angular momentum as a generator
    • Addition of two angular momenta
    • Summary and forward view
    • Problems
  12. Central potentials
    • Why central potentials
    • The Schrödinger equation in spherical coordinates
    • The effective potential and asymptotic behaviour
    • General features: nodes, degeneracy, and quantum numbers
    • The 3D infinite spherical well
    • The 3D isotropic harmonic oscillator
    • Preview: the Coulomb problem
    • Summary and forward view
    • Problems
  13. Spin-orbit coupling and the addition of angular momenta
    • Why spin-orbit coupling matters
    • Origin of spin-orbit coupling
    • Addition of orbital and spin angular momenta
    • Fine structure of the hydrogen atom
    • The anomalous Zeeman effect and the Landé g-factor
    • The nuclear shell model and the magic numbers
    • Summary and forward view
    • Problems
  14. The hydrogen atom
    • Why hydrogen
    • The Coulomb radial equation
    • The hydrogen wavefunctions
    • Probability densities and expectation values
    • Hidden symmetry: the Runge-Lenz vector and SO(4)
    • Fine structure and the Lamb shift
    • Summary and forward view
    • Problems
  15. Symmetries and conservation laws
    • Why symmetries matter
    • Continuous symmetries and conservation laws
    • Discrete symmetries: parity and time reversal
    • Hidden symmetries: Coulomb SO(4) and oscillator SU(3)
    • Selection rules from the Wigner-Eckart theorem
    • Gauge symmetry: a preview
    • Summary and forward view
    • Problems
  16. Time-independent perturbation theory
    • Why perturbation theory
    • Non-degenerate perturbation theory
    • Degenerate perturbation theory
    • The Stark effect on hydrogen
    • Hydrogen fine structure via perturbation theory
    • The Zeeman effect: weak, strong, and intermediate regimes
    • Summary and forward view
    • Problems
  17. Identical particles, many-body ground states, and intro to statistical mechanics
    • Why identical particles
    • The symmetrisation postulate
    • The helium atom
    • Slater determinants and multi-electron atoms
    • The periodic table and Hund’s rules
    • Bose-Einstein and Fermi-Dirac distributions
    • Summary and forward view
    • Problems
  18. Variational and WKB methods
    • Why variational and WKB
    • The variational principle
    • Variational calculations in practice: helium and H2+
    • The WKB approximation
    • Bound states: the Bohr-Sommerfeld quantisation rule
    • Tunnelling and the Gamow factor
    • Summary of Chapter 18
    • Problems
  19. Time-dependent perturbation theory and transition rates
    • Why time-dependent perturbation theory
    • The interaction picture and the Dyson series
    • Harmonic perturbations and the resonance structure
    • Fermi’s golden rule
    • Electric-dipole transitions and selection rules
    • Spontaneous emission and the Einstein coefficients
    • Summary and forward view
    • Problems
  20. Quantum information fundamentals
    • Why quantum information
    • Qubits and quantum gates
    • Quantum circuits
    • The no-cloning theorem
    • Quantum teleportation
    • Superdense coding
    • BB84 quantum key distribution
    • Summary and forward view
    • Problems
  21. Quantum algorithms
    • Why quantum algorithms matter
    • Quantum parallelism and interference
    • The Deutsch-Jozsa algorithm
    • Grover’s search algorithm
    • The quantum Fourier transform
    • Shor’s factoring algorithm
    • Complexity classes and quantum supremacy
    • Summary and forward view
    • Problems
  22. Quantum error correction and fault tolerance
    • Why error correction matters for quantum computing
    • Classical repetition codes
    • The 3-qubit bit-flip code
    • The 3-qubit phase-flip code
    • Shor’s 9-qubit code
    • The stabiliser formalism
    • The threshold theorem and fault-tolerant computation
    • The surface code
    • Summary and forward view
    • Problems
  23. Foundations and interpretations
    • The measurement problem
    • The Copenhagen interpretation
    • The many-worlds interpretation
    • De Broglie-Bohm pilot-wave theory
    • Consistent histories
    • Objective-collapse theories
    • QBism and epistemic interpretations
    • What is empirically at stake?
    • Summary and forward view
    • Problems
  24. Open quantum systems
    • Why open systems
    • Quantum channels: Kraus operators
    • The Lindblad master equation
    • Example: the damped harmonic oscillator
    • Decoherence and pointer states
    • Amplitude damping and phase damping: T1​ and T2​
    • Non-Markovian dynamics: a brief look
    • Summary and forward view
    • Problems
  25. Path integrals
    • Why path integrals
    • The propagator from time slicing
    • The free particle
    • Quadratic Lagrangians and the harmonic oscillator
    • The classical limit and stationary phase
    • Imaginary time: Wick rotation and statistical mechanics
    • Toward quantum field theory
    • Summary and forward view
    • Problems
  26. Quantisation of the electromagnetic field
    • Why quantise the field
    • Classical EM field in a cavity
    • Quantisation: creation and annihilation operators
    • The vacuum and its energy
    • Spontaneous emission rigorously derived
    • The Casimir effect
    • Toward quantum electrodynamics
    • Summary and forward view
    • Problems
  27. Relativistic wave equations and the problem of antiparticles
    • Why relativistic QM
    • Special relativity refresher
    • The Klein-Gordon equation
    • Problems with the one-particle interpretation
    • The Klein-Gordon field as a quantum field
    • Antiparticles and charge conjugation
    • Interactions
    • Summary and forward view
    • Problems
  28. The Dirac equation and quantum electrodynamics
    • Why the Dirac equation
    • Gamma matrices and the Clifford algebra
    • Free Dirac plane waves
    • Non-relativistic limit: the Pauli equation and g=2
    • Hydrogen from the Dirac equation
    • Dirac field quantisation and QED
    • Where this book ends, and what comes next
    • Summary of Chapter 28
    • Problems