# David Tong: Lectures on Applications of Quantum Mechanics

This is an advanced course on quantum mechanics. It covers a wide range of topics, including an introduction to condensed matter physics and scattering theory. Please do email me if you find any typos or mistakes.

PostScript
PDF

Cambridge students: The lecture notes on this page contain substantially more material than is needed for the
Part II course. A truncated set of notes that is closer to the syllabus
can be found here.

# Content

**1. Particles in Magnetic Fields: Postscript PDF**

Gauge field, gauge transformation; Landau levels, degeneracy; Aharonov-Bohm effect; Magnetic monopoles, Dirac quantisation; Spin in a magnetic field, spin precession.**2. Band Structure: Postscript PDF**

Electrons in one dimension, tight-binding, nearly free electrons, Floquet matrix, Bloch's theorem; Bravais lattices, cubic, BCC and FCC, the Wigner-Seitz cell, the reciprocal lattice, the Brillouin zone; band structure, crystal momentum, crysallographic notation, nearly free electrons in 3d, tight-binding in 3d; Wannier functions, localised and extended stats, LCAO**3. Electron Dynamics in Solids: Postscript PDF**

Fermi surfaces, metals vs insulators, graphene; Bloch electrons; effective velocity and mass, semi-classical equations of motion, Bloch oscillations, holes, Drude model; magnetic fields, cylcotron frequency, Onsager quantisation, de Haas-van Alphen oscillations.**4. Phonons: Postscript PDF**

Monotonic chain; diatomic chain, optical and accoustic bands, Peierls instability; Quantization; Field theory.**5. Discrete Symmetries: Postscript PDF**

Parity; Time Reversal, Kramers' degeneracy.**6. Approximation Methods: Postscript PDF**

The variational method; the helium atom; bound states, the Yukawa potential, the virial theorem; excited states. WKB, Semi-classical expansion, Linear potentials and the Airy function, Bohr-Sommerfeld quantisation, Tunnelling; The Sudden approximation, Quantum quenches; The Adiabatic approximation; Berry phase; The Born-Oppenheimer approximation, Molecular binding.**7. Atoms: Postscript PDF**

Hydrogen; Spin-Orbit coupling, Fine structure, Hyperfine structure; Helium, Exchange energy; Hartree method, Slater determinant, Hartree-Fock method.**8. Atoms in Electromagnetic Fields: Postscript PDF**

The Stark effect; The Zeeman effect; Rabi oscillations, Spontaneous emission, Selection rules, Photons, The Jaynes-Cummings model.**9. Quantum Foundations: Postscript PDF**

Entanglement, The EPR paradox, Bell's inquality, CHSH inequality, GHZ states, The Kochen-Specker theorem; Entanglement is a resource, The CHSH game, Dense coding, Quantum teleportation, Quantum key distribution; Density matrices, The Bloch sphere, Entropy; Projective measurements, Generalised measurements; Open quantum systems, Decoherence, The Lindblad equation.**10. Scattering Theory: Postscript PDF**

Scattering in one dimension, reflection and transmission coefficients, S-matrix, bound states, resonances; Scattering in three dimensions, the cross-section, the scattering amplitude, partial waves, phase shifts and the optical theorem, a hard sphere, bound states and resonances again; the Lippmann-Schwinger equation, the Born approximation, Yukawa and Coulomb potentials, the Born expansion; Rutherford scattering, the hydrogen atom; Scattering off a lattice, Bragg condition, structure factor, Debye-Waller factor.

# Problem Sheets

**Problem Sheet 1: Postscript PDF**Scattering**Problem Sheet 2: Postscript PDF**Variational Method, 1d Band Structure**Problem Sheet 3: Postscript PDF**3d Band Structure; Fermi Surfaces**Problem Sheet 4: Postscript PDF**Phonons; Particles in a Magnetic Field**Notes on Spherical Bessel Functions: Postscript PDF**

# Quantum Mechanics on the Web

**Applications of Quantum Mechanics**An earlier version of this course by Ron Horgan**Quantum Mechanics**by Robert Littlejohn at Berkeley**Advanced Quantum Mechanics**by Ben Simons in TCM, Cambridge**Solid State Physics**by Steve Simon at Oxford**Solid State Theory**by Manfred Sigrist at ETH, Zurich**Quantum Information**by John Preskill at Caltech