Choosing an Approximation Method

Choosing an approximation method is part of the physics. A method is not selected by the shape of an equation alone; it is selected by the observable, the limiting regime, the available exactly solved problem, and the diagnostic checks one can perform.

Use this page as a decision guide before reaching for formulas.

Step 1: Identify the Observable

Ask what you are trying to compute:

• an energy level,
• an eigenstate correction,
• an expectation value,
• a transition probability,
• a transition rate,
• a tunneling exponent,
• a scattering amplitude,
• a cross section,
• an effective low-energy Hamiltonian,
• a classical-limit phase.

The same Hamiltonian may require different methods for different observables. For example, a weak potential can be treated perturbatively for bound-state energy shifts and through the Born approximation for scattering amplitudes.

Step 2: Find the Solvable Core

Many approximations begin by splitting

$$H=H_0+V.$$

The split is useful only if $H_0$ is solved well enough for the observable of interest. That usually means knowing eigenstates, eigenvalues, propagators, or Green functions.

If no useful $H_0$ exists, a variational method or numerical method may be more natural than formal perturbation theory.

Step 3: Check for Degeneracy

If a perturbation mixes states with nearby or equal unperturbed energies, ordinary nondegenerate formulas can fail. The diagnostic ratio is often

$$\frac{\lvert V_{mn}\rvert}{\lvert E_m^{(0)}-E_n^{(0)}\rvert}.$$

When this ratio is not small for states in the relevant subspace, diagonalize the perturbation inside that subspace or use an effective Hamiltonian.

Degeneracy is not a small technicality. It changes the starting point of the approximation.

Step 4: Decide Whether Time Matters

If the Hamiltonian depends explicitly on time,

$$H(t)=H_0+V(t),$$

then energy shifts are usually not the first question. Ask instead whether the drive is weak, sudden, slow, periodic, or resonant.

• Weak drive: use time-dependent perturbation theory.
• Dense continuum of final states: use a rate formula such as Fermi’s golden rule.
• Slow change with gaps: use the adiabatic approximation.
• Abrupt change: use the sudden approximation.
• Fast periodic drive: use an effective or Magnus-type expansion.

The boundary between these regimes is set by time scales and energy gaps.

Step 5: Look for a Variational Bound

If the target is a ground-state energy and exact solution is hard, the variational principle is often the cleanest first estimate:

$$E_0 \le \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}.$$

This gives a one-sided bound, not merely an uncontrolled guess, provided the Hamiltonian and trial state satisfy the usual assumptions.

Variational methods are especially useful when a physically motivated trial state captures the main length scale or symmetry of the problem.

Step 6: Ask Whether the Limit Is Semiclassical

WKB and semiclassical methods are natural when phases vary rapidly:

$$\psi(x)\sim A(x)e^{iS(x)/\hbar}.$$

In one dimension, define

$$p(x)=\sqrt{2m(E-V(x))}.$$

WKB is plausible when the local de Broglie wavelength changes slowly. A common diagnostic is

$$\left| \frac{d}{dx}\frac{\hbar}{p(x)} \right| \ll 1,$$

away from turning points.

Near turning points, ordinary WKB fails and connection formulas are needed.

Step 7: Recognize Scattering Problems

If the physical setup involves an incoming beam, outgoing flux, and asymptotic states, it is a scattering problem. The natural observables are not bound-state energies but amplitudes and cross sections.

For short-range potentials in three dimensions, the asymptotic form is commonly written

$$\psi(\mathbf r) \sim e^{ikz} + f(\theta,\phi)\frac{e^{ikr}}{r}.$$

Then

$$\frac{d\sigma}{d\Omega} = \lvert f(\theta,\phi)\rvert^2.$$

Weak potentials suggest the Born approximation. Central potentials suggest partial waves and phase shifts. Low-energy scattering often requires scattering length rather than a naive weak-potential expansion.

Step 8: Use Symmetry Before Calculating

Symmetry can show that a correction vanishes, that a perturbation block-diagonalizes, or that only certain transitions are allowed. Before calculating, check conservation laws, parity, angular momentum, and selection rules.

This is not optional polish. Symmetry often decides which approximation is valid.

A Compact Decision List

Use perturbation theory when a solved Hamiltonian is nearby and mixing is small.

Use degenerate perturbation theory when states of equal or nearby unperturbed energy are coupled.

Use time-dependent perturbation theory when a weak drive causes transitions.

Use a variational method when estimating a ground-state energy from a trial family.

Use WKB when the action is large compared with $\hbar$ and the potential varies slowly.

Use scattering theory when the observable is an outgoing flux or cross section.

Use effective Hamiltonians when the problem has separated subspaces or energy scales.

Common Mistakes

• Starting with a memorized formula before identifying the observable.
• Using nondegenerate perturbation theory inside a degenerate subspace.
• Calling a parameter small without making it dimensionless.
• Applying WKB at a turning point without a connection formula.
• Using the Born approximation for a strong or resonant scattering potential.
• Ignoring symmetry constraints that make a leading correction vanish.

Cross-Links

• Volume Overview
• Approximation Map
• Small Parameters and Error Estimates
• Perturbative, Variational, and Asymptotic Thinking
• Common Failure Modes
• Instantons, Tunneling, and Nonperturbative Effects
• Time-Dependent Hamiltonians
• Sudden Approximation
• Adiabatic Approximation as a Method
• Symmetry Constraints on Hamiltonians

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• 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.
• A. Messiah, Quantum Mechanics, Dover, 1999.

Exercises

1. A perturbation connects two unperturbed states with matrix element $10^{-3}\,\mathrm{eV}$ and energy separation $10^{-5}\,\mathrm{eV}$. Should ordinary nondegenerate perturbation theory be trusted for those two states?

Solution

The mixing ratio is

$$\frac{10^{-3}\,\mathrm{eV}}{10^{-5}\,\mathrm{eV}} = 100.$$

This is not small. Nondegenerate perturbation theory should not be trusted for that pair of states; one should diagonalize within the relevant near-degenerate subspace or use an effective two-state model.

2. A particle scatters from a weak short-range potential. Which quantity should be computed first: an energy correction or a scattering amplitude?

Solution

For a scattering setup the observable is outgoing flux, usually encoded in a scattering amplitude $f(\theta,\phi)$ and cross section. A weak short-range potential suggests the Born approximation for the scattering amplitude rather than a bound-state energy correction.