Approximation Map

Approximation methods are not a pile of unrelated tricks. They are organized by the structure that controls the calculation: a small coupling, a separated subspace, a variational bound, a large action, a weak drive, or asymptotic boundary conditions.

This page maps the main families before the technical pages begin.

The Central Split

Most approximation problems start from one of three situations:

• You know a solvable Hamiltonian $H_0$ and add a correction.
• You know a useful trial family of states and optimize within it.
• You know a limiting regime, such as large action, weak scattering, or slow variation.

These situations produce different kinds of trust. A perturbative answer is trusted because powers of a small parameter are suppressed. A variational answer is trusted because it bounds an energy under appropriate assumptions. A semiclassical answer is trusted because an asymptotic phase varies rapidly compared with $\hbar$.

Method Families

Perturbation theory starts with

$$H(\lambda)=H_0+\lambda V.$$

It expands energies, states, or transition amplitudes in powers of a parameter $\lambda$. The expansion is controlled when the perturbation weakly mixes the states of interest compared with relevant energy gaps.

Variational methods choose trial states $\lvert\psi(\alpha)\rangle$ and minimize

$$E[\psi] = \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}.$$

For ground states, this gives an upper bound under the standard self-adjoint Hamiltonian assumptions.

WKB and semiclassical methods use a phase of the form

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

They are controlled when the action changes on scales large compared with $\hbar$, or equivalently when the potential varies slowly on the local de Broglie wavelength scale.

Transition theory studies amplitudes between initial and final states under a time-dependent perturbation. It is controlled by weak driving, finite-time phases, and the density of final states.

Scattering theory replaces bound-state normalization with asymptotic boundary conditions. The core observables are amplitudes, cross sections, phase shifts, and unitarity constraints.

Effective Hamiltonian methods eliminate or fold away degrees of freedom when energy scales or time scales are separated.

Exact, Approximate, and Asymptotic

An exact identity is true within its assumptions without a small parameter. A controlled approximation is not exact, but it states why neglected terms are small. An asymptotic approximation may improve for a while as more terms are added, even if the full series does not converge.

Perturbation series in quantum mechanics are often asymptotic rather than convergent. This is not a defect by itself. It means the series must be used with the right error expectations.

Perturbative and Nonperturbative

Perturbative methods expand around a chosen solvable limit. Nonperturbative effects are invisible to every finite order in that expansion. Tunneling splittings and instanton contributions are standard examples.

A common pattern is

$$\text{perturbative terms} \sim 1+\lambda+\lambda^2+\cdots,$$

while a tunneling contribution may scale like

$$e^{-S/\hbar}.$$

No finite power series in $\hbar$ reproduces that exponential dependence.

Energy-Domain and Time-Domain Thinking

Some methods are naturally spectral:

• time-independent perturbation theory,
• variational bounds,
• WKB quantization,
• phase shifts in stationary scattering.

Others are naturally dynamical:

• time-dependent perturbation theory,
• sudden and adiabatic approximations,
• Landau-Zener transitions,
• Magnus and Floquet-Magnus expansions.

The distinction is not absolute. Scattering, for example, can be described through stationary states or time-dependent wave packets. But choosing the natural viewpoint usually reduces algebraic clutter.

Observable-First Map

If the desired output is an energy shift, start with perturbation theory, degenerate perturbation theory, or a variational estimate.

If the output is a transition probability or rate, start with time-dependent perturbation theory.

If the output is a tunneling exponent, first identify whether the observable is transmission, a level splitting, or a decay rate; the nonperturbative method map then directs the calculation to WKB, an instanton, or a bounce.

If the output is a cross section, start with scattering amplitude conventions.

If the output is an effective low-energy description, start with projection or scale-separation methods.

If the output is a classical-limit approximation, start with WKB, stationary phase, or semiclassical propagators.

Where Boundaries Matter

This volume owns the quantum-mechanical methods. Detailed molecular spectroscopy, quantum-matter band topology, open-system master equations, and relativistic scattering each have their own canonical homes. The method pages here should point outward rather than duplicate those applications.

Cross-Links

• Volume Overview
• Volume Landing Page
• Choosing a Method
• Small Parameters and Error Estimates
• Perturbative, Variational, and Asymptotic Thinking
• Instantons, Tunneling, and Nonperturbative Effects
• Which Formulation Should I Use?
• Symmetry, Angular Momentum, and Spin

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.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315-397, 1972.

Exercises

1. Classify each approximation as perturbative, variational, semiclassical, scattering-based, or effective-Hamiltonian based: Rayleigh-Ritz, WKB tunneling, Born approximation, Schrieffer-Wolff transformation, and first-order energy shifts.

Solution

Rayleigh-Ritz is variational. WKB tunneling is semiclassical. Born approximation is scattering-based and perturbative in the potential. Schrieffer-Wolff is an effective-Hamiltonian method based on scale separation. First-order energy shifts are time-independent perturbation theory.

2. Give an example of a nonperturbative effect and explain why a power series misses it.

Solution

A tunneling splitting often scales as $e^{-S/\hbar}$. Expanding in powers of $\hbar$ around $\hbar=0$ cannot reproduce this behavior at any finite order. It is exponentially small, not polynomially small.