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.

One-Page Route

1. If the target is a bound-state energy or eigenstate, ask whether the Hamiltonian is close to a solved Hamiltonian.
2. If the level is isolated, use Nondegenerate Perturbation Theory.
3. If the level is degenerate or nearly degenerate, use Degenerate Perturbation Theory or an effective subspace method.
4. If no small perturbation is available but a good trial state is available, use the Variational Principle or Rayleigh-Ritz Method.
5. If the target is a transition probability or rate under a weak time-dependent interaction, use First-Order Transition Probability or Fermi’s Golden Rule.
6. If the wavefunction has a slowly varying local wavelength or the action is large compared with $\hbar$, use WKB Approximation.
7. If the problem has smooth classical turning points, include Turning Points and Connection Formulas.
8. If the target is a smooth-well spectrum, use Bohr-Sommerfeld Quantization.
9. If the target is a tunneling exponent through a smooth barrier, use Barrier Penetration and Tunneling.
10. If the setup is an incoming beam and outgoing flux, use scattering theory: Scattering Amplitude and Differential and Total Cross Sections.
11. If the scattering potential is weak and short-ranged, try the First Born Approximation.
12. If the scattering potential is central, use Partial-Wave Expansion and Phase Shifts.
13. If the result is a scattering amplitude, check unitarity with the Optical Theorem when applicable.

Diagnostic Questions

What is the output?

• Energy shift: perturbation theory, variational methods, or numerical diagonalization.
• Eigenstate correction: perturbation theory or finite-basis methods.
• Transition probability: time-dependent perturbation theory.
• Transition rate: golden-rule limit.
• Bound-state spectrum at large quantum number: WKB quantization.
• Tunneling probability: WKB tunneling or later instanton methods.
• Scattering angle distribution: scattering amplitude and cross section.
• Central-potential scattering data: phase shifts.

What is small or large?

• Small perturbing potential compared with energy gaps: perturbation theory.
• Small drive matrix element over relevant timescale: transition perturbation theory.
• Large action compared with $\hbar$: WKB or stationary phase.
• Large under-barrier action: tunneling exponent.
• Weak short-range scattering potential: Born approximation.
• Large basis with stable convergence: Rayleigh-Ritz.
• Large separation between low- and high-energy subspaces: effective Hamiltonian methods.

What can go singular?

• Energy denominators can become small.
• Degenerate states can mix before shifting.
• WKB can fail at turning points.
• Golden-rule probabilities can outgrow the perturbative regime at long times.
• Born scattering can fail near resonances or shallow bound states.
• Cross-section formulas can change when channels, spin, or identical particles matter.

Common Routes

For a perturbed oscillator:

1. Use Nondegenerate Perturbation Theory for weak anharmonicity.
2. Use the Variational Principle for a ground-state upper bound.
3. Use Rayleigh-Ritz Method for numerical benchmarking.
4. Use WKB only for high-lying states or semiclassical questions.

For a weak driven atom:

1. Identify the perturbing interaction.
2. Compute the matrix element.
3. Use First-Order Transition Probability for finite-time amplitudes.
4. Use Fermi’s Golden Rule only when the final states form a continuum or dense spectrum.

For a short-range central scattering potential:

1. Define the scattering amplitude convention.
2. If the potential is weak, compute the Born amplitude.
3. If low energy or strong scattering matters, compute phase shifts.
4. Use cross sections as flux observables.
5. Check the optical theorem when the approximations should respect unitarity.

Stop Conditions

Stop and reconsider the method if:

• the small parameter is not dimensionless;
• a denominator is comparable to the perturbation;
• a supposedly small transition probability is no longer small;
• WKB is being used directly at a turning point;
• a variational result is being interpreted as a lower bound;
• a scattering approximation violates basic flux conservation without explanation;
• a page or calculation cannot state its validity regime in words.

After choosing a method, use the Error-Estimate Checklist to audit the result before reporting it.

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 check cross sections and unitarity limits.

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

• 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.
• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.