PU-401 · Advanced Quantum Theory
Building on introductory quantum mechanics, this unit develops the machinery a working quantum physicist actually reaches for: time-dependent perturbation theory and radiative transitions, the full apparatus of angular-momentum coupling and symmetry, semiclassical and variational approximations, and a rigorous treatment of scattering. It closes by widening the framework itself — geometric phase, identical-particle statistics, mixed states, and the path-integral formulation — so the student sees quantum theory as a single interlocking structure rather than a bag of tricks.
Lectures
| L01 | Recap: Postulates, Pictures, and the Structure of the Theory — |
| L02 | The Interaction Picture and the Dyson Series |
| L03 | Transitions to a Continuum: Fermi's Golden Rule |
| L04 | Atoms in Radiation Fields: the Semiclassical Treatment |
| L05 | Selection Rules, Parity, and Spectral Lines |
| L06 | Spontaneous Emission and the Einstein Coefficients |
| L07 | Angular Momentum Revisited: Coupling Two Systems |
| L08 | Clebsch-Gordan Coefficients and the Coupled Basis |
| L09 | Tensor Operators and the Wigner-Eckart Theorem |
| L10 | Approximation Methods I: the Variational Principle |
| L11 | Approximation Methods II: the WKB Method |
| L12 | Connection Formulas and Semiclassical Quantization |
| L13 | Tunneling and Alpha Decay: the Gamow Factor |
| L14 | Slow Change: the Adiabatic Theorem |
| L15 | Berry's Geometric Phase |
| L16 | Introduction to Scattering: Cross Sections and Amplitudes — |
| L17 | The Lippmann-Schwinger Equation |
| L18 | The Born Approximation and its Range of Validity |
| L19 | Partial Waves and Phase Shifts |
| L20 | The Optical Theorem and Unitarity |
| L21 | Identical Particles and Exchange Symmetry |
| L22 | Mixed States and the Density Matrix |
| L23 | Entanglement and Reduced Density Matrices |
| L24 | The Feynman Path Integral |
| L25 | Path Integrals: Free Particle and the Classical Limit |
| L26 | Synthesis: Symmetry, Approximation, and the Unity of the Framework — |
Derivations homed in this unit
Interaction Picture and the Dyson Series
Derives the time-ordered Dyson-series expansion for the evolution operator of a Hamiltonian split into a solvable part plus a time-dependent perturbation.
Fermi's Golden Rule
Derives the transition rate to a continuum of final states from the long-time limit of the first-order transition probability.
Semiclassical Radiation and Dipole Selection Rules
Treats the atom-field interaction in the dipole approximation and derives the electric-dipole selection rules on the angular-momentum and parity quantum numbers.
Einstein A and B Coefficients
Derives the fixed ratios between spontaneous emission, stimulated emission, and absorption rates by demanding detailed balance against the Planck spectrum.
Addition of Angular Momenta
Decomposes the tensor product of two angular-momentum representations into irreducible sectors, constructing the coupled basis via ladder operators and orthogonality.
The Wigner-Eckart Theorem
Proves that matrix elements of a spherical tensor operator factor into a geometric Clebsch-Gordan coefficient and a single reduced matrix element.
The Rayleigh-Ritz Variational Bound
Proves that the expectation value of the Hamiltonian in any trial state is an upper bound on the true ground-state energy.
WKB Approximation and Connection Formulas
Derives the semiclassical wavefunction from the ħ-expansion of the phase and matches oscillatory to exponential regions across a classical turning point via the Airy solution.
Gamow Tunneling and Alpha Decay
Derives the exponential barrier-penetration factor and the resulting decay-rate scaling from the WKB transmission integral.
The Adiabatic Theorem
Proves that a system prepared in a non-degenerate eigenstate remains in the instantaneous eigenstate when the Hamiltonian varies slowly compared with the level spacing.
Berry's Geometric Phase
Derives the geometric phase accumulated over a closed adiabatic loop in parameter space as the flux of the Berry curvature.
The Lippmann-Schwinger Equation
Recasts the scattering Schrödinger equation as an integral equation using the outgoing free Green's function.
The Born Approximation
Derives the first-order scattering amplitude as the Fourier transform of the potential and the resulting differential cross section.
Partial-Wave Analysis and Phase Shifts
Expands the scattering amplitude in Legendre partial waves and expresses each in terms of a real phase shift extracted from the asymptotic radial solution.
The Optical Theorem
Derives that the total cross section is proportional to the imaginary part of the forward scattering amplitude, expressing conservation of probability flux.
Identical Particles and Exchange Symmetry
Derives the symmetrization postulate, Slater determinants for fermions, and the exchange contribution to energy.
The Density Matrix and von Neumann Equation
Constructs the density operator for mixed states, derives its equation of motion, and obtains the reduced density matrix by partial trace over a subsystem.
The Feynman Path Integral
Derives the propagator as a sum over paths weighted by exp(iS/ħ) from time-slicing the evolution operator, recovering classical mechanics in the stationary-phase limit.