Method Comparison Table

Use this table to choose a method after identifying the observable, the available exact starting point, and the small or large dimensionless parameter. Common Symbols provides a compact notation key for the entries below.

Method	Problem Type	Input	Output	Control Parameter	Strengths	Failure Modes	Canonical Examples	Related Volumes
Nondegenerate perturbation	Isolated energy level	Solved Hamiltonian plus weak correction	Energy and state corrections	Perturbation divided by level gaps	Direct, systematic, good for spectroscopy	Small denominators, degeneracy, divergent asymptotics	Weak anharmonicity	Core formalism, AMO
Degenerate perturbation	Degenerate multiplet	Perturbation matrix in degenerate subspace	Splittings and good zeroth-order states	Gap to states outside subspace	Handles symmetry-protected degeneracy	Wrong basis, neglected near-degenerate states	Stark and Zeeman-type splittings	Symmetry, spin, AMO
First-order transitions	Weak time-dependent drive	Interaction-picture matrix element	Transition amplitude and probability	Drive strength times duration	Shows resonance and phase matching	Long-time breakdown, strong driving	Harmonic perturbations	Dynamics, AMO
Fermi’s golden rule	Transition into continuum	Matrix element and density of states	Rate	Weak coupling and continuum limit	Converts amplitudes into rates	Discrete final states, strong coupling, short times	Decays and absorption	QFT bridge, open systems
Variational principle	Ground-state estimate	Trial state family	Upper bound on ground energy	Trial-space quality	Works without small coupling	Rigid trial family, bad observables	Gaussian trial states	Computational QM
Rayleigh-Ritz	Finite-basis eigenproblem	Basis and matrix elements	Ritz eigenvalues and vectors	Basis size and conditioning	Systematic numerical route	Truncation artifacts, ill-conditioned overlap	Oscillator-basis diagonalization	Numerical methods
WKB	Slowly varying one-dimensional potential	Local momentum	Semiclassical wavefunction	Wavelength changes slowly	Gives phases and asymptotics	Turning points, abrupt jumps	Smooth wells	Canonical systems
WKB tunneling	Smooth barrier	Under-barrier action	Transmission exponent	Barrier action large compared with hbar	Robust exponential estimates	Prefactor errors, near-top barriers	Field emission, alpha decay	Quantum matter, QFT bridge
First Born approximation	Weak short-range scattering	Fourier transform of potential	Scattering amplitude	Weak potential or high energy	Simple link between angle and potential structure	Strong scattering, resonances, Coulomb tails	Gaussian and Yukawa potentials	QFT bridge
Partial waves	Central-potential scattering	Radial equations by angular momentum	Phase shifts and amplitude	Convergent angular-momentum sum	Enforces rotational symmetry and unitarity	Noncentral potentials, long-range subtleties	Low-energy scattering	Symmetry, rigorous QM
Optical theorem	Scattering consistency check	Forward amplitude	Total cross section constraint	Unitarity	Detects probability-conservation failures	Convention mistakes, missing channels	Partial-wave proof	QFT unitarity

How to Read the Table

First choose the row by observable, not by preference. Energy shifts, transition rates, wavefunctions, tunneling exponents, and cross sections are different outputs. A method that is excellent for one output may be the wrong language for another.

Second identify the control parameter. It may be a small coupling, a large action, a large basis, a weak drive, a large energy gap, or a symmetry restriction. If there is no plausible control parameter, state that explicitly and compare against numerical or exact limiting cases.

Third check the failure modes before calculating. Most serious approximation errors come from using the right formula in the wrong regime.

Before reporting the result, run the Error-Estimate Checklist.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.