Validity and Error Control
An approximation controls a specified prediction within a specified regime. Its reliability depends on the observable, the parameters held fixed in the limit, and the size or character of the omitted contribution. This chapter distinguishes a useful small parameter from an error bound and explains how a failed check can reveal which part of a calculation needs to change.
Helpful background. States and operators supply the Hamiltonian and expectation-value language; asymptotic analysis defines order symbols and finite remainders.
Control parameters compare physical scales
Section titled “Control parameters compare physical scales”A dimensional coupling is not small by itself. For mixing between two levels, compare a coupling matrix element with their energy separation. For a driven system, both the coupling and the elapsed time can enter the control parameter. For a semiclassical wavefunction, local wavelength variation matters even when the total action is large.
The anharmonic oscillator illustrates the first step: divide the Hamiltonian by a natural energy and express position in a natural length. The resulting dimensionless coupling identifies a limit, but the useful range can still depend on the level number. A weak-coupling statement at fixed low excitation need not hold uniformly across the spectrum.
Errors belong to observables
Section titled “Errors belong to observables”Write a finite approximation as
The claim about must match the accuracy needed for . A bound on an absolute energy need not resolve a much smaller level splitting. A variational energy bound does not automatically bound a wavefunction tail or a transition matrix element. Relative error is especially unhelpful near a zero unless a meaningful nonzero comparison scale is supplied.
An omitted term can suggest an error scale, but a rigorous remainder inequality makes a stronger statement. Conservation of norm or flux is a necessary consistency check, not a numerical error bar. Agreement under one numerical refinement tests that source of error; it does not eliminate model, boundary, or other truncation errors.
State where the approximation is uniform
Section titled “State where the approximation is uniform”An asymptotic remainder at fixed time, distance, or level number need not remain small when that variable grows as the control parameter decreases. A weak resonant drive can accumulate an order-one transition; the Rabi example compares the short-time expansion with the full oscillation. Near a turning point, the local semiclassical expansion requires a different solution and matching.
A useful validity statement therefore names the parameter limit, the retained order, and the region or time interval over which the estimate applies. When those conditions fail, a smaller expansion parameter alone may not repair the calculation: the starting subspace, local solution, or time evolution may need to be reorganized.
Develop and test the error estimate
Section titled “Develop and test the error estimate”| Question | Detailed treatment |
|---|---|
| What distinguishes a small ratio, an error estimate, and a bound? | Small Parameters and Error Estimates derives an exact two-level remainder bound. |
| What does each approximation principle establish? | Perturbative, Variational, and Asymptotic Thinking compares coefficients, trial-state bounds, and fixed-order remainders. |
| Which assumption failed, and what should change? | Common Failure Modes connects small denominators, secular growth, turning points, and other symptoms to quantitative checks and repairs. |
Use the error-estimate checklist when stating the result of a calculation, and notation and conventions when comparing its symbols with another source.
References
Section titled “References”- C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, 1999. Nonuniform limits and reorganized approximations.
- F. W. J. Olver and R. Wong, “Definitions and Elementary Properties,” §2.1, in NIST Digital Library of Mathematical Functions, Chapter 2: Asymptotic Approximations. Order symbols, remainders, and uniformity.
- R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994. Perturbative and variational methods and their physical assumptions.