Quantum Physics III

Perturbation theory, the semiclassical limit, scattering and identical particles.

Syllabus

  1. Lectures

    1. L1.1 General problem. Non-degenerate perturbation theory
    2. L1.2 Setting up the perturbative equations
    3. L1.3 Calculating the energy corrections
    4. L1.4 First order correction to the state. Second order correction to energy
    5. L2.1 Remarks and validity of the perturbation series
    6. L2.2 Anharmonic Oscillator via a quartic perturbation
    7. L2.3 Degenerate Perturbation theory: Example and setup
    8. L2.4 Degenerate Perturbation Theory: Leading energy corrections
    9. L3.1 Remarks on a 'good basis'
    10. L3.2 Degeneracy resolved to first order; state and energy corrections
    11. L3.3 Degeneracy resolved to second order
    12. L3.4 Degeneracy resolved to second order (continued)
    13. L4.1 Scales and zeroth-order spectrum
    14. L4.2 The uncoupled and coupled basis states for the spectrum
    15. L4.3 The Pauli equation for the electron in an electromagnetic field
    16. L4.4 Dirac equation for the electron and hydrogen Hamiltonian
    17. L5.1 Evaluating the Darwin correction
    18. L5.2 Interpretation of the Darwin correction from nonlocality
    19. L5.3 The relativistic correction
    20. L5.4 Spin-orbit correction
    21. L5.5 Assembling the fine-structure corrections
    22. L6.1 Zeeman effect and fine structure
    23. L6.2 Weak-field Zeeman effect; general structure
    24. L6.3 Weak-field Zeeman effect; the projection lemma
    25. L6.4 Strong-field Zeeman
    26. L6.5 Semiclassical approximation and local de Broglie wavelength
    27. L7.1 The WKB approximation scheme
    28. L7.2 Approximate WKB solutions
    29. L7.3 Validity of the WKB approximation
    30. L7.4 Connection formula stated and example
    31. L8.1 Airy functions as integrals in the complex plane
    32. L8.2 Asymptotic expansions of Airy functions
    33. L8.3 Deriving the connection formulae
    34. L8.4 Deriving the connection formulae (continued) logical arrows
    35. L9.1 The interaction picture and time evolution
    36. L9.2 The interaction picture equation in an orthonormal basis
    37. L9.3 Example: Instantaneous transitions in a two-level system
    38. L9.4 Setting up perturbation theory
    39. L10.1 Box regularization: density of states for the continuum
    40. L10.2 Transitions with a constant perturbation
    41. L10.3 Integrating over the continuum to find Fermi's Golden Rule
    42. L10.4 Autoionization transitions
    43. L11.1 Harmonic transitions between discrete states
    44. L11.2 Transition rates for stimulated emission and absorption processes
    45. L11.3 Ionization of hydrogen: conditions of validity, initial and final states
    46. L11.4 Ionization of hydrogen: matrix element for transition
    47. L12.1 Ionization rate for hydrogen: final result
    48. L12.2 Light and atoms with two levels, qualitative analysis
    49. L12.3 Einstein's argument: the need for spontaneous emission
    50. L12.4 Einstein's argument: B and A coefficients
    51. L12.5 Atom-light interactions: dipole operator
    52. L13.1 Transition rates induced by thermal radiation
    53. L13.2 Transition rates induced by thermal radiation (continued)
    54. L13.3 Einstein's B and A coefficients determined. Lifetimes and selection rules
    55. L13.4 Charged particles in EM fields: potentials and gauge invariance
    56. L13.5 Charged particles in EM fields: Schrodinger equation
    57. L14.1 Gauge invariance of the Schrödinger Equation
    58. L14.2 Quantization of the magnetic field on a torus
    59. L14.3 Particle in a constant magnetic field: Landau levels
    60. L14.4 Landau levels (continued). Finite sample
    61. L15.1 Classical analog: oscillator with slowly varying frequency
    62. L15.2 Classical adiabatic invariant
    63. L15.3 Phase space and intuition for quantum adiabatic invariants
    64. L15.4 Instantaneous energy eigenstates and Schrodinger equation
    65. L16.1 Quantum adiabatic theorem stated
    66. L16.2 Analysis with an orthonormal basis of instantaneous energy eigenstates
    67. L16.3 Error in the adiabatic approximation
    68. L16.4 Landau-Zener transitions
    69. L16.5 Landau-Zener transitions (continued)
    70. L17.1 Configuration space for Hamiltonians
    71. L17.2 Berry's phase and Berry's connection
    72. L17.3 Properties of Berry's phase
    73. L17.4 Molecules and energy scales
    74. L18.1 Born-Oppenheimer approximation: Hamiltonian and electronic states
    75. L18.2 Effective nuclear Hamiltonian. Electronic Berry connection
    76. L18.3 Example: The hydrogen molecule ion
    77. L19.1 Elastic scattering defined and assumptions
    78. L19.2 Energy eigenstates: incident and outgoing waves. Scattering amplitude
    79. L19.3 Differential and total cross section
    80. L19.4 Differential as a sum of partial waves
    81. L20.1 Review of scattering concepts developed so far
    82. L20.2 The one-dimensional analogy for phase shifts
    83. L20.3 Scattering amplitude in terms of phase shifts
    84. L20.4 Cross section in terms of partial cross sections. Optical theorem
    85. L20.5 Identification of phase shifts. Example: hard sphere
    86. L21.1 General computation of the phase shifts
    87. L21.2 Phase shifts and impact parameter
    88. L21.3 Integral equation for scattering and Green's function
    89. L22.1 Setting up the Born Series
    90. L22.2 First Born Approximation. Calculation of the scattering amplitude
    91. L22.3 Diagrammatic representation of the Born series. Scattering amplitude for spherically symm...
    92. L22.4 Identical particles and exchange degeneracy
    93. L23.1 Permutation operators and projectors for two particles
    94. L23.2 Permutation operators acting on operators
    95. L23.3 Permutation operators on N particles and transpositions
    96. L23.4 Symmetric and Antisymmetric states of N particles
    97. L24.1 Symmetrizer and antisymmetrizer for N particles
    98. L24.2 Symmetrizer and antisymmetrizer for N particles (continued)
    99. L24.3 The symmetrization postulate
    100. L24.4 The symmetrization postulate (continued)