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.


PostScript      PDF


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.

Content


  • 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