Foundational Quantum Mechanics

Derive the angular momentum commutation relations from the algebraic properties of infinitesimal rotations.

Determine the eigenvalue spectrum of J² and Jz algebraically using the ladder operator method.

Construct spherical harmonics as simultaneous eigenfunctions of L² and Lz and interpret their nodal structure.

Represent spin-1/2 using Pauli matrices, derive the SU(2) rotation matrices for spinors, and contrast spinor and vector transformation properties.

Derive the Clebsch-Gordan series for the addition of two angular momenta and determine the allowed total angular momentum quantum numbers.

Compute Clebsch-Gordan coefficients for simple systems (e.g., coupling two spin-1/2 particles) and construct coupled basis states.

Define irreducible tensor operators and apply the Wigner-Eckart theorem to evaluate matrix elements and extract selection rules.

Apply time-reversal symmetry to quantum systems and derive Kramers' degeneracy for half-integer spin systems.

Solve the quantum harmonic oscillator using the algebraic (ladder operator) method and derive the energy eigenvalue spectrum.

Construct normalized energy eigenstates of the harmonic oscillator and compute matrix elements of position and momentum using ladder operators.

Construct coherent states as eigenstates of the annihilation operator and show they are minimum-uncertainty states that follow classical trajectories.

Separate the two-body central force problem into center-of-mass and relative coordinates.

Solve the radial Schrödinger equation for the Coulomb potential to derive the hydrogen atom's energy spectrum and radial wave functions.

Explain the n²-fold degeneracy of the hydrogen atom and relate the additional degeneracy beyond rotational symmetry to the conserved Laplace-Runge-Lenz vector.

Compute expectation values of powers of r for hydrogen eigenstates using the Hellmann-Feynman theorem or Kramers' recursion relation.

Derive and interpret the probability current density and its continuity equation.

Compute reflection and transmission coefficients for scattering from a potential step and a rectangular barrier.

Analyze quantum tunneling through a potential barrier, deriving the transmission coefficient and its exponential dependence on barrier parameters.

Solve for bound and scattering states of the Dirac delta function potential as a model for short-range interactions.

Define the scattering amplitude and differential cross-section in three dimensions.

Develop the partial wave analysis method for spherically symmetric potentials.

Derive the Born approximation for the scattering amplitude and establish its conditions of validity.

Apply the Born approximation to calculate the differential cross-section for the Yukawa potential.

State the symmetrization postulate and classify particles as bosons or fermions based on the spin-statistics connection.

Construct properly symmetrized and anti-symmetrized multi-particle states for systems of identical particles.

Write anti-symmetrized fermionic states as Slater determinants and derive the Pauli exclusion principle as a consequence.

Explain the physical origin of the exchange interaction and distinguish it from a direct force.

Analyze the singlet-triplet splitting in two-electron systems and relate it to the symmetry of spatial and spin wave functions.

Analyze the Stern-Gerlach experiment as a demonstration of spin quantization and the projection postulate.

Solve for the dynamics of a spin-1/2 particle in a uniform magnetic field and derive Larmor precession.

Analyze spin dynamics in an oscillating magnetic field to derive the condition for magnetic resonance.

Derive the spin-orbit coupling Hamiltonian as a relativistic correction using the Thomas precession argument.

Derive the first and second-order energy corrections and the first-order state correction for a non-degenerate system.

Formulate degenerate perturbation theory and apply it to lift degeneracies by diagonalizing the perturbation within the degenerate subspace.

Calculate the relativistic kinetic energy and Darwin term corrections to the hydrogen spectrum using first-order perturbation theory.

Compute the spin-orbit correction to the hydrogen spectrum, leading to the fine-structure formula.

Compute the first-order Stark effect in hydrogen using degenerate perturbation theory and determine the resulting energy splittings.

Derive the weak-field Zeeman effect using first-order perturbation theory and compute the Landé g-factor.

Analyze the strong-field (Paschen-Back) regime of the Zeeman effect and contrast it with the weak-field limit.

State and prove the variational principle as a method for obtaining an upper bound on the ground state energy.

Apply the variational method with a parameterized trial wave function to estimate the ground-state energy of an anharmonic oscillator.

Estimate the ground-state energy of the helium atom using both first-order perturbation theory and the variational method, and compare the results.

Derive the WKB ansatz from the Schrödinger equation and establish its regime of validity.

Apply the WKB connection formulas at classical turning points to match solutions across classically allowed and forbidden regions.

Derive the Bohr-Sommerfeld quantization condition and apply it to compute approximate energy spectra.

Estimate tunneling rates through arbitrary potential barriers using the WKB approximation.

Derive the time-dependent perturbation theory framework and obtain the transition amplitude to first order.

Compute first-order transition probabilities for constant and sinusoidal perturbations, identifying resonance behaviour.

Derive Fermi's golden rule for transition rates into a continuum of final states.

Apply Fermi's golden rule and the electric dipole approximation to derive the rates of absorption and stimulated emission (Einstein's B coefficients).

Analyze the electric dipole matrix element to derive the selection rules for atomic transitions.

State the adiabatic theorem and explain its conditions of validity.

Compute the Berry phase for a spin-1/2 particle in a slowly rotating magnetic field and explain its geometric origin.

Contrast the adiabatic and sudden approximations and determine which regime applies for given time scales.