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L2.4 Degenerate Perturbation Theory: Leading energy corrections
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Classroom Contents
Quantum Physics III (Spring 2018)
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- 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 Schrodinger 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)
