quantum-mechanics

name	quantum_mechanics
description	Use when solving quantum mechanical problems including the Schrodinger equation, angular momentum coupling, scattering theory, or many-body quantum systems.

Quantum Mechanics

Apply this skill for problems involving quantum states, operators, time evolution, measurement, angular momentum algebra, scattering amplitudes, or many-body quantum systems.

Goal

Set up and solve quantum mechanical problems using the Schrodinger equation, operator algebra, symmetry principles, and standard approximation methods.

Scope

• Time-independent Schrodinger equation: bound states, energy spectra, wavefunctions
• Time-dependent Schrodinger equation: evolution operators, transition amplitudes
• Angular momentum: addition of angular momenta, Clebsch-Gordan coefficients, Wigner-Eckart theorem
• Identical particles: bosons, fermions, Slater determinants, second quantization
• Scattering theory: Born approximation, partial wave analysis, optical theorem, S-matrix
• Density matrices and mixed states
• WKB approximation for semiclassical problems
• Path integral formulation (when appropriate)

Inputs

• hamiltonian: The Hamiltonian operator (or potential for single-particle problems)
• initial_state: Initial wavefunction or quantum state
• observables: Operators whose expectation values or spectra are sought
• boundary_conditions: Normalizability, periodicity, scattering boundary conditions

Outputs

• energy_spectrum: Eigenvalues and their degeneracies
• wavefunctions: Eigenstates or time-evolved states
• expectation_values: for requested observables
• transition_amplitudes: Matrix elements, scattering amplitudes, cross sections

Workflow

1. Formulate the Hamiltonian and identify the Hilbert space.

   • Choose a suitable basis (position, momentum, energy, angular momentum).
   • Identify symmetries to simplify the problem (parity, rotational symmetry -> good quantum numbers).

2. Solve the eigenvalue problem (time-independent case).

   • For exactly solvable potentials (harmonic oscillator, hydrogen atom, infinite well): use analytic methods.
   • For general potentials: use variational methods, perturbation theory, or numerical diagonalization.
   • Apply boundary conditions: normalizability for bound states, asymptotic plane waves for scattering.

3. Time evolution (time-dependent case).

   • For time-independent H: psi(t) = exp(-iHt/h-bar) psi(0), expand in energy eigenstates.
   • For time-dependent H: use time-dependent perturbation theory or numerical propagation.

4. Compute observables.

   • = <psi|O|psi> for pure states; Tr(rho O) for mixed states.
   • Uncertainties: delta O = sqrt(<O^2> - ^2).
   • Transition rates: Fermi's golden rule for perturbative transitions.

5. Scattering (if applicable).

   • Set up the asymptotic boundary conditions: incident plane wave + outgoing spherical wave.
   • Compute the scattering amplitude f(theta, phi).
   • Differential cross section: d sigma / d Omega = |f|^2.
   • Total cross section: sigma_tot via the optical theorem.

Quality Checks

• Wavefunctions must be normalizable (bound states) or have proper scattering asymptotics.
• Energy eigenvalues must be real for Hermitian Hamiltonians.
• Uncertainty relations must be satisfied: delta x * delta p >= h-bar / 2.
• Unitarity: S^dag S = 1; probability is conserved.
• For angular momentum coupling, verify that total J quantum numbers satisfy the triangle inequality.

Constraints

• Do not confuse state vectors with wavefunctions; keep the representation explicit.
• For identical particles, always impose the correct exchange symmetry (antisymmetric for fermions, symmetric for bosons).
• WKB is valid only when the potential varies slowly compared to the local de Broglie wavelength; do not apply it near classical turning points without connection formulas.
• The Born approximation is valid for weak scattering potentials; do not use it for strong potentials without justification.