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.

Methods by Observable and Available Scale

Method and target	What you need	What controls accuracy	Main limitation
Nondegenerate perturbation: isolated energies and states	solved reference spectrum and interaction matrix elements	coupling-to-gap ratios and the combined omitted-state norm	many small mixings can accumulate; high-order series may diverge
Degenerate perturbation: a level cluster	full interaction inside the cluster	coupling out of the cluster divided by external gaps	a missing nearby state invalidates the chosen cluster
First-order transitions: finite-time probabilities	interaction-picture drive and initial state	integrated amplitude; on resonance roughly interaction × time / $\hbar$	depletion and coherent saturation
Golden rule: continuum rate	on-shell matrix element and density of states	weak coupling and $\tau_{\mathrm{corr}}\ll t\ll\Gamma^{-1}$	discrete recurrences and spectral thresholds
Variational method: ground-energy bound	normalized trial family in the form domain	trial-space flexibility and independent residuals	a good energy need not imply accurate observables
Rayleigh–Ritz: discrete low spectrum	exactly projected Hamiltonian and overlap	nested basis enlargement and full-space residuals	boundary construction errors or ill-conditioned basis
WKB: phase and high-level spectrum	local momentum and boundary conditions	large orbit action and slow local variation	turning points need matching; low states can be inaccurate
Tunneling: barrier exponent	forbidden-region action	large action divided by $\hbar$	accurate exponent does not guarantee relative accuracy of a probability or gap
Effective Hamiltonian: low-energy dynamics	retained cluster, external gap, and couplings	coupling/gap, time range, and transformed observables	resonant eliminated states and long-time phase accumulation
Magnus expansion: propagator	time-dependent Hamiltonian and commutators	convergence bounds or a controlled asymptotic regime	a unitary approximation can still have large phase error
First Born: weak scattering	potential and incident energy	repeated-scattering corrections for that range and energy	threshold resonances and long-range tails

Partial-wave expansion and the optical theorem are exact structures under their stated scattering assumptions, not small-parameter approximations. A partial-wave cutoff is an approximation and must be increased until the observable stabilizes; see the partial-wave discussion.

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 I: Asymptotic Methods and Perturbation Theory, Springer, 1999.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.