Worked Problems and Model Calculations

These worked problems show how an approximation answers a particular physical question. The same potential can call for different methods: weak-coupling perturbation theory for an energy coefficient, a variational calculation for an upper bound, or WKB for highly excited levels. Choose a problem by the quantity you want to understand, then compare the approximation with an exact limit or a converged numerical solution.

Helpful background. Approximation decision tree connects observables with useful approaches. Small parameters and errors explains what an accuracy claim means.

Start with two views of one oscillator

The quartic oscillator is a good first comparison. In perturbation theory, calculate how a weak quartic term shifts the harmonic spectrum. In the variational calculation, adjust a Gaussian width to lower the energy. The answers address the same ground state, but their guarantees differ: the perturbation series gives asymptotic coefficients, while the exact expectation in an admissible trial state gives an upper bound.

Read the problem statement before the solution and identify the energy and length scales yourself. Then reproduce one matrix element or width derivative without looking. Finally compare the regimes of validity. Agreement at weak coupling does not mean that the two methods remain equally accurate as the coupling grows.

Perturbation theory benchmarks extends this comparison to basis convergence. A stable diagonalization is valuable only after the basis is large enough for the particular levels being compared.

Choose a calculation

Problem	What you learn to predict	A useful check
Two-level Stark shift	Field-induced mixing and the quadratic energy response	Expand the exact two-state eigenvalues; assess omitted levels separately.
Landau–Zener crossing	Transfer through a swept avoided crossing	Keep adiabatic and diabatic labels distinct and test finite sweep endpoints.
Smooth-well WKB levels	The action-quantized pure-quartic spectrum	Check units and compare low and high levels with numerical eigenvalues.
WKB barrier tunneling	Leading exponential suppression	Compare the opaque rectangular-barrier limit and identify the missing interface prefactor.
Double-well splitting	The small gap between even and odd states	Resolve both symmetry sectors and compare the splitting, not only their mean energy.

Together with the two anharmonic-oscillator calculations, these form the seven worked articles in this chapter. The method pages linked within each example supply the reusable derivations.

For collisions, continue with the scattering worked problems. The delta potential is a compact exercise in matching and flux; the spherical square well develops partial-wave phases; the hard sphere separates low-energy waves from high-energy diffraction.

The Gaussian and Yukawa examples test the first Born approximation. Low-energy s-wave scattering explores range corrections near threshold, while a square-well resonance distinguishes a cross-section maximum from a complex pole. These links are useful comparisons after the bound-state and transition problems above.

How to check your answer

First check dimensions and exact structure: parity, normalization, Hermiticity, and probability conservation often reveal errors before a numerical comparison. Then test a solvable limit, such as zero perturbation or a known oscillator spectrum.

Next ask whether the comparison is independent of the approximation. Two quadrature routines evaluating the same Born integral test the integral’s numerical evaluation; they do not test the neglect of repeated scattering. A radial Schrödinger solution or an exact scattering model tests that physical approximation more directly.

Finally compare the intended observable over the intended range. A good energy does not guarantee a good transition matrix element. A correct tunneling exponent does not guarantee a precise probability. The error-estimate guide and computational examples develop these distinctions further.

Exercises

A matched exponent. A tunneling approximation has the correct leading exponential but predicts twice the exact prefactor. Is it asymptotically accurate in relative error?

Solution

No. If $Q_{\mathrm{app}}/Q\to2$ , the relative error tends to one. The approximation captures the leading exponential dependence, which can be useful across many orders of magnitude, but it does not give relative accuracy. State explicitly which level of accuracy is intended.

A changing reference. At a fixed coupling, a perturbative energy differs from a numerical result by $0.002$ in chosen units. Increasing the numerical basis changes that result by $0.005$ . What should you do before judging perturbation theory?