Applications of Quantum Mechanics

Applications of Quantum Mechanics(D)

APPLICATIONS OF QUANTUM MECHANICS (D)
Principles of Quantum Mechanics is essential.

Variational Principle

  • Variational principle, examples. [2]

Bound states and scattering states in one dimension

  • Bound states, reflection and transmission amplitudes. Examples. Relation between bound states and transmission amplitude by analytic continuation. [3]

Scattering theory in three dimensions

  • Classical scattering, definition of differential cross section. Asymptotic wavefunction for quantum scattering, scattering amplitude, cross section. Green’s function, Born approximation to scattering on a potential. Spherically symmetric potential, partial waves and phase shifts, optical theorem. Low energy scattering, scattering length. Bound states and resonances as zeros and poles of S-matrix. [5]

Electrons in a magnetic field

  • Vector potential and Hamiltonian. Quantum Hamiltonian, inclusion of electron spin, gauge invariance, Zeeman splitting. Landau levels, effect of spin, degeneracy and filling effects, use of complex variable for lowest Landau level. Aharonov-Bohm effect. [4]

Particle in a one-dimensional periodic potential

  • Discrete translation group, lattice and reciprocal lattice, periodic functions. Bloch’s theorem, Brillouin zone, energy bands and gaps. Floquet matrix, eigenvalues. Band gap in nearly-free electron model, tight-binding approximation. [3]

Crystalline solids

  • Introduction to crystal symmetry groups in three dimensions, Voronoi/Wigner-Seitz cell. Primitive, body-centred and face-centred cubic lattices. Reciprocal lattice, periodic functions, lattice planes, Brillouin zone. Bloch states, electron bands, Fermi surface. Basics of electrical conductivity: insulators, semiconductors, conductors. Extended zone scheme. [4]
  • Bragg scattering. Vibrations of crystal lattice, quantization, phonons. [3]

Appropriate books

  • D.J. Griffiths Introduction to Quantum Mechanics. 2nd edition, Pearson Education 2005
  • R. Shankar Principles of Quantum Mechanics. Springer 2008
  • N.W. Ashcroft and N.D. Mermin Solid State Physics. Holt–Saunders 1976

Associated GitHub page

https://jaircambridge.github.io/Applications-of-Quantum-Mechanics-D/