| 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 stateobservables: Operators whose expectation values or spectra are soughtboundary_conditions: Normalizability, periodicity, scattering boundary conditions
Outputs
energy_spectrum: Eigenvalues and their degeneracieswavefunctions: Eigenstates or time-evolved statesexpectation_values: for requested observablestransition_amplitudes: Matrix elements, scattering amplitudes, cross sections
Workflow
-
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).
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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.
-
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.
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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.
-
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.
