Approximation Decision Tree

The fastest way to choose an approximation is to start with the observable. Do not begin by asking which method is familiar. Ask what must be computed, what is known exactly, and what parameter controls the error.

Helpful background. Small parameters and error estimates explains how dimensionless ratios and the target observable determine the accuracy check.

Choose the Observable First

An energy or stationary state

First use symmetry constraints to identify invariant subspaces and forbidden matrix elements.

If a solved reference problem is available, compare the interaction matrix elements with the gaps. Small mixing of an isolated level suggests nondegenerate perturbation theory. Many individually small mixing amplitudes can still produce a large total state correction; test their combined norm as in small parameters and error estimates. A degenerate or nearly degenerate cluster must first be diagonalized together; test its coupling to the remaining states separately.

Without a small coupling, a physically motivated trial state gives a variational ground-energy bound. A growing Rayleigh–Ritz basis can improve it. For highly excited states in a smooth one-dimensional well, large action instead suggests WKB quantization, with the appropriate turning-point phases.

A transition or time evolution

For a weak drive over a time short enough to keep transition probabilities small, use time-dependent perturbation theory. A smooth dense continuum may turn the probability into a golden-rule rate over times long compared with its correlation time but short enough that depletion and recurrences remain negligible. A single coherently driven resonant level instead exhibits Rabi oscillations; a density-of-states formula does not replace that dynamics.

For slow variation, examine gaps and derivative couplings before using the adiabatic approximation. For periodic or rapidly varying drives, Magnus or a justified high-frequency expansion may be useful. Check the full time interval and the observable, not just a frequency ratio.

For a change short compared with relevant dynamical times, the sudden approximation keeps the state nearly unchanged during the switch. A large integrated impulse can invalidate that conclusion even when the switch is brief.

A tunneling effect

A thick barrier suggests an exponential of the forbidden-region action. First decide whether the target is a transmission probability, a closed-well splitting, or metastable decay. Their exponents, prefactors, and boundary conditions differ. The double-well example shows why a splitting is linear in a tunneling amplitude rather than its square.

A scattering observable

Define incoming and outgoing flux and the amplitude convention. For weak short-range scattering, examine the Born approximation. For a central potential, partial waves separate angular momentum at any strength; only the radial solution or angular-momentum cutoff introduces an approximation. Near a threshold resonance, phase shifts or the effective-range expansion are usually more useful than first Born.

Small incident momentum alone does not imply weak scattering. An unscreened Coulomb interaction requires long-range asymptotic states, so the short-range starting assumptions must also change.

Check the optical theorem at the order being retained. A real first Born forward amplitude has zero imaginary part, while its cross section is second order in the potential; that imaginary part appears at second Born order. This mismatch alone is not a failure of perturbation theory.

When to Change the Starting Point

Reconsider the approximation if an omitted state becomes resonant, the total transition probability becomes order one, a turning point is crossed without matching, or increasing the numerical resolution changes the answer. A large relative error near a zero may instead require a different error scale. The Error-Estimate Checklist explains these distinctions, and the comparison table summarizes each method’s inputs.

Practice Checks

1. A particle scatters from a weak Gaussian potential at moderate energy. Which method comes first?

Solution

The observable is a scattering amplitude or cross section, and the potential is weak and short-ranged. Start with the first Born approximation, then estimate higher Born terms or compare with radial solutions. Apply unitarity consistently across perturbative orders.

2. A bound state belongs to a two-dimensional degenerate eigenspace of $H_0$. Which method should be used first?

Solution

Use degenerate perturbation theory. Diagonalize the perturbation inside the degenerate subspace before applying nondegenerate formulas to the resulting good zeroth-order states.

3. A smooth potential well has highly excited bound states. Which approximation is natural?

Solution

Use WKB and Bohr-Sommerfeld quantization, provided the turning points are handled with connection formulas and the action is large compared with $\hbar$.

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.