Quantum Physics III (Spring 2018)

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This course is a continuation of 8.05 Quantum Physics II. It introduces some of the important model systems studied in contemporary physics, including two-dimensional electron systems, the fine structure of hydrogen, lasers, and particle scattering.

An edX version of this course,8.06x Applications of Quantum Mechanics, is available starting on February 20, 2019 and running for 19 weeks.

L1.1 General problem. Non-degenerate perturbation theory.
L1.2 Setting up the perturbative equations.
L1.3 Calculating the energy corrections.
L1.4 First order correction to the state. Second order correction to energy.
L2.1 Remarks and validity of the perturbation series.
L2.2 Anharmonic Oscillator via a quartic perturbation.
L2.3 Degenerate Perturbation theory: Example and setup.
L2.4 Degenerate Perturbation Theory: Leading energy corrections.
L3.1 Remarks on a 'good basis'.
L3.2 Degeneracy resolved to first order; state and energy corrections.
L3.3 Degeneracy resolved to second order.
L3.4 Degeneracy resolved to second order (continued).
L4.1 Scales and zeroth-order spectrum.
L4.2 The uncoupled and coupled basis states for the spectrum.
L4.3 The Pauli equation for the electron in an electromagnetic field.
L4.4 Dirac equation for the electron and hydrogen Hamiltonian.
L5.1 Evaluating the Darwin correction.
L5.2 Interpretation of the Darwin correction from nonlocality.
L5.3 The relativistic correction.
L5.4 Spin-orbit correction.
L5.5 Assembling the fine-structure corrections.
L6.1 Zeeman effect and fine structure.
L6.2 Weak-field Zeeman effect; general structure.
L6.3 Weak-field Zeeman effect; the projection lemma.
L6.4 Strong-field Zeeman.
L6.5 Semiclassical approximation and local de Broglie wavelength.
L7.1 The WKB approximation scheme.
L7.2 Approximate WKB solutions.
L7.3 Validity of the WKB approximation.
L7.4 Connection formula stated and example.
L8.1 Airy functions as integrals in the complex plane.
L8.2 Asymptotic expansions of Airy functions.
L8.3 Deriving the connection formulae.
L8.4 Deriving the connection formulae (continued) logical arrows.
L9.1 The interaction picture and time evolution.
L9.2 The interaction picture equation in an orthonormal basis.
L9.3 Example: Instantaneous transitions in a two-level system.
L9.4 Setting up perturbation theory.
L10.1 Box regularization: density of states for the continuum.
L10.2 Transitions with a constant perturbation.
L10.3 Integrating over the continuum to find Fermi's Golden Rule.
L10.4 Autoionization transitions.
L11.1 Harmonic transitions between discrete states.
L11.2 Transition rates for stimulated emission and absorption processes.
L11.3 Ionization of hydrogen: conditions of validity, initial and final states.
L11.4 Ionization of hydrogen: matrix element for transition.
L12.1 Ionization rate for hydrogen: final result.
L12.2 Light and atoms with two levels, qualitative analysis.
L12.3 Einstein's argument: the need for spontaneous emission.
L12.4 Einstein's argument: B and A coefficients.
L12.5 Atom-light interactions: dipole operator.
L13.1 Transition rates induced by thermal radiation.
L13.2 Transition rates induced by thermal radiation (continued).
L13.3 Einstein's B and A coefficients determined. Lifetimes and selection rules.
L13.4 Charged particles in EM fields: potentials and gauge invariance.
L13.5 Charged particles in EM fields: Schrodinger equation.
L14.1 Gauge invariance of the Schrodinger Equation.
L14.2 Quantization of the magnetic field on a torus.
L14.3 Particle in a constant magnetic field: Landau levels.
L14.4 Landau levels (continued). Finite sample.
L15.1 Classical analog: oscillator with slowly varying frequency.
L15.2 Classical adiabatic invariant.
L15.3 Phase space and intuition for quantum adiabatic invariants.
L15.4 Instantaneous energy eigenstates and Schrodinger equation.
L16.1 Quantum adiabatic theorem stated.
L16.2 Analysis with an orthonormal basis of instantaneous energy eigenstates.
L16.3 Error in the adiabatic approximation.
L16.4 Landau-Zener transitions.
L16.5 Landau-Zener transitions (continued).
L17.1 Configuration space for Hamiltonians.
L17.2 Berry's phase and Berry's connection.
L17.3 Properties of Berry's phase.
L17.4 Molecules and energy scales.
L18.1 Born-Oppenheimer approximation: Hamiltonian and electronic states.
L18.2 Effective nuclear Hamiltonian. Electronic Berry connection.
L18.3 Example: The hydrogen molecule ion.
L19.1 Elastic scattering defined and assumptions.
L19.2 Energy eigenstates: incident and outgoing waves. Scattering amplitude.
L19.3 Differential and total cross section.
L19.4 Differential as a sum of partial waves.
L20.1 Review of scattering concepts developed so far.
L20.2 The one-dimensional analogy for phase shifts.
L20.3 Scattering amplitude in terms of phase shifts.
L20.4 Cross section in terms of partial cross sections. Optical theorem.
L20.5 Identification of phase shifts. Example: hard sphere.
L21.1 General computation of the phase shifts.
L21.2 Phase shifts and impact parameter.
L21.3 Integral equation for scattering and Green's function.
L22.1 Setting up the Born Series.
L22.2 First Born Approximation. Calculation of the scattering amplitude.
L22.3 Diagrammatic representation of the Born series. Scattering amplitude for spherically symm....
L22.4 Identical particles and exchange degeneracy.
L23.1 Permutation operators and projectors for two particles.
L23.2 Permutation operators acting on operators.
L23.3 Permutation operators on N particles and transpositions.
L23.4 Symmetric and Antisymmetric states of N particles.
L24.1 Symmetrizer and antisymmetrizer for N particles.
L24.2 Symmetrizer and antisymmetrizer for N particles (continued).
L24.3 The symmetrization postulate.
L24.4 The symmetrization postulate (continued).