Quantum Physics III

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L1.1 General problem. Non-degenerate perturbation theory

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L1.2 Setting up the perturbative equations

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L1.3 Calculating the energy corrections

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L1.4 First order correction to the state. Second order correction to energy

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L2.1 Remarks and validity of the perturbation series

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L2.2 Anharmonic Oscillator via a quartic perturbation

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L2.3 Degenerate Perturbation theory: Example and setup

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L2.4 Degenerate Perturbation Theory: Leading energy corrections

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L3.1 Remarks on a 'good basis'

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L3.2 Degeneracy resolved to first order; state and energy corrections

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L3.3 Degeneracy resolved to second order

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L3.4 Degeneracy resolved to second order (continued)

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L4.1 Scales and zeroth-order spectrum

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L4.2 The uncoupled and coupled basis states for the spectrum

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L4.3 The Pauli equation for the electron in an electromagnetic field

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L4.4 Dirac equation for the electron and hydrogen Hamiltonian

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L5.1 Evaluating the Darwin correction

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L5.2 Interpretation of the Darwin correction from nonlocality

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L5.3 The relativistic correction

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L5.4 Spin-orbit correction

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L5.5 Assembling the fine-structure corrections

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L6.1 Zeeman effect and fine structure

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L6.2 Weak-field Zeeman effect; general structure

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L6.3 Weak-field Zeeman effect; the projection lemma

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L6.4 Strong-field Zeeman

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L6.5 Semiclassical approximation and local de Broglie wavelength

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L7.1 The WKB approximation scheme

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L7.2 Approximate WKB solutions

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L7.3 Validity of the WKB approximation

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L7.4 Connection formula stated and example

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L8.1 Airy functions as integrals in the complex plane

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L8.2 Asymptotic expansions of Airy functions

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L8.3 Deriving the connection formulae

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L8.4 Deriving the connection formulae (continued) logical arrows

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L9.1 The interaction picture and time evolution

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L9.2 The interaction picture equation in an orthonormal basis

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L9.3 Example: Instantaneous transitions in a two-level system

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L9.4 Setting up perturbation theory

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L10.1 Box regularization: density of states for the continuum

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L10.2 Transitions with a constant perturbation

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L10.3 Integrating over the continuum to find Fermi's Golden Rule

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L10.4 Autoionization transitions

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L11.1 Harmonic transitions between discrete states

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L11.2 Transition rates for stimulated emission and absorption processes

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L11.3 Ionization of hydrogen: conditions of validity, initial and final states

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L11.4 Ionization of hydrogen: matrix element for transition

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L12.1 Ionization rate for hydrogen: final result

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L12.2 Light and atoms with two levels, qualitative analysis

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L12.3 Einstein's argument: the need for spontaneous emission

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L12.4 Einstein's argument: B and A coefficients

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L12.5 Atom-light interactions: dipole operator

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L13.1 Transition rates induced by thermal radiation

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L13.2 Transition rates induced by thermal radiation (continued)

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L13.3 Einstein's B and A coefficients determined. Lifetimes and selection rules

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L13.4 Charged particles in EM fields: potentials and gauge invariance

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L13.5 Charged particles in EM fields: Schrodinger equation

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L14.1 Gauge invariance of the Schrödinger Equation

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L14.2 Quantization of the magnetic field on a torus

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L14.3 Particle in a constant magnetic field: Landau levels

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L14.4 Landau levels (continued). Finite sample

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L15.1 Classical analog: oscillator with slowly varying frequency

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L15.2 Classical adiabatic invariant

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L15.3 Phase space and intuition for quantum adiabatic invariants

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L15.4 Instantaneous energy eigenstates and Schrodinger equation

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L16.1 Quantum adiabatic theorem stated

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L16.2 Analysis with an orthonormal basis of instantaneous energy eigenstates

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L16.3 Error in the adiabatic approximation

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L16.4 Landau-Zener transitions

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L16.5 Landau-Zener transitions (continued)

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L17.1 Configuration space for Hamiltonians

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L17.2 Berry's phase and Berry's connection

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L17.3 Properties of Berry's phase

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L17.4 Molecules and energy scales

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L18.1 Born-Oppenheimer approximation: Hamiltonian and electronic states

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L18.2 Effective nuclear Hamiltonian. Electronic Berry connection

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L18.3 Example: The hydrogen molecule ion

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L19.1 Elastic scattering defined and assumptions

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L19.2 Energy eigenstates: incident and outgoing waves. Scattering amplitude

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L19.3 Differential and total cross section

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L19.4 Differential as a sum of partial waves

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L20.1 Review of scattering concepts developed so far

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L20.2 The one-dimensional analogy for phase shifts

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L20.3 Scattering amplitude in terms of phase shifts

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L20.4 Cross section in terms of partial cross sections. Optical theorem

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L20.5 Identification of phase shifts. Example: hard sphere

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L21.1 General computation of the phase shifts

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L21.2 Phase shifts and impact parameter

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L21.3 Integral equation for scattering and Green's function

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L22.1 Setting up the Born Series

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L22.2 First Born Approximation. Calculation of the scattering amplitude

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L22.3 Diagrammatic representation of the Born series. Scattering amplitude for spherically symm...

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L22.4 Identical particles and exchange degeneracy

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L23.1 Permutation operators and projectors for two particles

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L23.2 Permutation operators acting on operators

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L23.3 Permutation operators on N particles and transpositions

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L23.4 Symmetric and Antisymmetric states of N particles

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L24.1 Symmetrizer and antisymmetrizer for N particles

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L24.2 Symmetrizer and antisymmetrizer for N particles (continued)

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L24.3 The symmetrization postulate

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L24.4 The symmetrization postulate (continued)