David Tong: Applications of Quantum Mechanics

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.

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These are the lecture notes given to final year undergraduates at the University of Cambridge. An extended version of this course can be found here.


  • 1. The Variational Method:   Postscript    PDF
    The variational method; the helium atom; bound states, the Yukawa potential, the virial theorem; excited states.
  • 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. 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.
  • 6. 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    Variational Method, Band Structure

  • Problem Sheet 2:   Postscript    PDF    More band structure, Fermi surfaces, Semi-classical dynamics, Phonons

  • Problem Sheet 3:   Postscript    PDF    Particles in magnetic fields, Scattering in 1d.   

  • Problem Sheet 4:   Postscript    PDF    Scattering in 3d

  • Notes on Spherical Bessel Functions:   Postscript    PDF

Quantum Mechanics on the Web