Error-Estimate Checklist

An approximation is useful when you understand both the answer and what it leaves out. This page explains how to assess a calculated energy, rate, or tunneling probability: choose a meaningful error scale, test sensitive regions, and distinguish a proved bound from a numerical indication. Small Parameters and Error Estimates develops the underlying ideas.

What an Error Estimate Is

Write the exact target quantity, when it exists, as

$$Q_{\mathrm{exact}} = Q_{\mathrm{approx}} + R.$$

The remainder $R$ is the approximation error. Different calculations provide different kinds of information about it:

Statement	Meaning	Appropriate Language
$\vert R\vert\le B$ under stated assumptions	rigorous error bound	“The error is bounded by $B$.”
$R=O(\epsilon^{N+1})$ as $\epsilon\to0$	asymptotic order at fixed $N$	“The omitted terms begin at order $\epsilon^{N+1}$.”
the first omitted term has size $a$	truncation estimate	“The next term suggests an error of scale $a$.”
two converged methods differ by $a$	benchmark discrepancy	“The observed discrepancy is $a$.”
a residual or conservation defect is small	error indicator	“The diagnostic is consistent with accuracy.”

These statements are not interchangeable. A small next term is not automatically a bound, and agreement with one benchmark is not a proof of uniform accuracy.

Relative error also requires care. The familiar ratio

$$\varepsilon_{\mathrm{rel}} = \frac{|R|}{|Q_{\mathrm{exact}}|}$$

is uninformative when the exact quantity vanishes or crosses zero. Near a node, selection-rule zero, or cancellation, report an absolute error relative to a physically chosen scale $Q_{\mathrm{ref}}$ instead:

$$\varepsilon_{\mathrm{scaled}} = \frac{|R|}{Q_{\mathrm{ref}}}.$$

Questions That Reveal Missing Physics

Use the questions relevant to your observable. A small energy error, for example, need not mean that a transition probability or a wavefunction tail is accurate.

1. Identify the dimensionless parameter

Name the parameter that is small or large. A dimensional coupling is not a control parameter until it is compared with the scale that competes with it.

Typical examples are

$$\epsilon_{\mathrm{PT}} = \max_{m\ne n} \frac{|\lambda V_{mn}|} {|E_n^{(0)}-E_m^{(0)}|},$$ $$\epsilon_{\mathrm{WKB}}(x) = \left| \frac{\hbar p'(x)}{p(x)^2} \right|,$$

The maximum mixing ratio is only a first check: many individually small terms can add to a substantial state correction. In intermediate normalization,

$$\bigl\|\lambda|n^{(1)}\rangle\bigr\|^2 =\sum_{m\ne n}\frac{|\lambda V_{mn}|^2}{|E_n^{(0)}-E_m^{(0)}|^2}.$$

Continuum contributions require integrals and domain/convergence assumptions. In scattering, both interaction strength and range relative to wavelength matter. There may be several control parameters. State which one governs each observable and whether the limit is uniform over the domain of interest.

2. Check energy denominators

List every denominator that can become small. For stationary perturbation theory, a useful diagnostic is

$$\Delta_{\min} = \min_{m\in\mathcal C} |E_n^{(0)}-E_m^{(0)}|,$$

where $\mathcal C$ contains states coupled appreciably to the state of interest. In driven systems, detunings play the same role. In resolvents and scattering, poles and thresholds replace discrete denominators.

Do not inspect only the nearest level in energy. A more distant state with a much larger matrix element can dominate the correction.

3. Check degeneracies and near degeneracies

An exact degeneracy is not merely a small numerical denominator. It changes the zeroth-order problem. Identify the relevant subspace $P$, diagonalize the perturbation or effective Hamiltonian inside it, and only then expand in coupling to the complementary subspace $Q$.

Near degeneracy requires a scale comparison. If a splitting $\Delta$ is comparable to an off-diagonal coupling $v$, then

$$\frac{|v|}{|\Delta|} \not\ll 1,$$

and nondegenerate formulas are not controlled even when $v$ is small in absolute units.

4. Check boundary and initial conditions

Verify that the approximate solution satisfies the physical problem, not just the local differential equation.

• Bound states must be normalizable and obey the required endpoint conditions.
• Scattering states must use the intended incoming and outgoing convention.
• Retarded, advanced, and time-ordered Green functions solve different boundary-value problems.
• WKB branches must be connected across turning regions rather than extended through singular formulas.
• Decay and resonance states require boundary conditions different from ordinary bound states.

A locally accurate expression with the wrong boundary condition describes a different observable.

5. Check units and normalization

Every retained term in a sum must have the same dimensions. Also check the dimensions of the final observable: amplitudes, rates, widths, cross sections, and probability densities are not interchangeable.

Normalization conventions matter whenever states belong to a continuum. For example, the dimensions assigned separately to a matrix element and a density of states can change, while the product in a physical rate remains invariant. Common Symbols and the Scattering Convention Dictionary provide quick translation aids.

6. Check symmetry constraints

List the exact symmetries of the Hamiltonian, state, and observable. Then test whether the approximation preserves their consequences:

• conserved quantum numbers,
• degeneracy patterns,
• parity and exchange symmetry,
• selection rules,
• Hermiticity or the appropriate non-Hermitian structure,
• gauge-independent observables,
• unitarity to the order being retained.

If symmetry forces the nominal leading term to vanish, the first nonzero contribution occurs at higher order. That changes both the estimate and the correct relative-error scale.

7. Compare with limiting cases

Take every limit whose answer is known before inserting numerical values. Useful checks include

$$\begin{gathered} \lambda\to0, \qquad \hbar\to0, \\ k\to0, \qquad k\to\infty. \end{gathered}$$

as well as infinite separation, equal masses, vanishing detuning, or restoration of a symmetry.

A limiting case can reveal a wrong sign, missing factor, incorrect branch, or nonuniform expansion. Passing a known limit is useful, but does not establish accuracy away from it.

8. Compare with exact or numerical benchmarks

Choose a benchmark that is independent of the approximation under test. Examples include exact diagonalization in a demonstrably converged basis, numerical integration of the original differential equation, a solvable model, or experimental data whose systematic uncertainties are understood.

For a sequence $Q_N$ produced by increasing a basis size, grid resolution, or expansion order, record at least

$$\Delta_N = |Q_N-Q_{N-1}|.$$

A decreasing $\Delta_N$ suggests numerical convergence for the chosen model. It does not test whether that model includes the interactions needed to describe the experiment.

9. Estimate omitted terms

Write the first omitted structure explicitly whenever possible. If

$$Q_N(\epsilon) = \sum_{r=0}^{N} a_r\epsilon^r,$$

then inspect $a_{N+1}\epsilon^{N+1}$ and the trend of neighboring terms. For an asymptotic series, the smallest retained term may be a better practical indicator than the next formal power after the terms begin growing.

Also identify effects outside the expansion itself: excluded channels, neglected operators, finite-volume corrections, truncation of a basis, or uncertainty in input parameters. Algebraic truncation error is only one part of the error budget.

10. State the regime of validity in words

End the calculation with a sentence that a reader can test. A useful pattern is:

The result retains terms through order $N$ in the parameter $\epsilon$, applies away from the stated singular region, and is expected to receive corrections of the indicated scale provided the listed gap, boundary, and symmetry conditions hold.

Replace generic phrases with the actual observable and scales. “Weak coupling” alone is not a validity statement.

Method-Specific Diagnostics

No single scalar diagnostic works for every approximation. Use the checks that match the method and observable.

Method	Primary Control	Useful comparison	Warning Sign
Nondegenerate perturbation theory	matrix elements over relevant gaps	next-order shift, exact diagonalization	small denominator or strong state mixing
Degenerate or effective-subspace method	coupling out of retained subspace over external gap	stability under enlarging the subspace	omitted state approaches the retained cluster
Time-dependent perturbation theory	transition amplitude and drive strength over the observation time	higher-order amplitude or exact few-level evolution	perturbative probability becomes order unity
Golden-rule rate	weak coupling plus an intermediate time window	finite-time calculation and stable continuum limit	discrete recurrences or substantial depletion
Variational method	quality and flexibility of trial space	monotone energy improvement, residual, independent observables	good energy but unstable wavefunction-sensitive quantities
Rayleigh–Ritz	basis completeness and conditioning	basis enlargement, residual norm	spectral pollution or ill-conditioned overlap matrix
WKB	slowly varying local wavelength and large action	exact/numerical matching, next semiclassical order	turning point, abrupt potential, or small action
Born scattering	weak repeated scattering in the relevant channel	second Born term, phase shifts, flux checks	resonance, long-range tail, or large unitarity defect
Partial-wave truncation	angular-momentum cutoff beyond contributing impact parameters	stability under increasing $\ell_{\max}$	long-range interaction or unresolved forward peak

Series truncation

Suppose the magnitudes of successive terms behave approximately geometrically over the computed orders,

$$\left| \frac{a_{r+1}\epsilon^{r+1}} {a_r\epsilon^r} \right| \lesssim q<1.$$

If that behavior can be justified for the remaining series, then a geometric-tail estimate gives

$$|R_N| \lesssim \frac{|a_{N+1}\epsilon^{N+1}|} {1-q}.$$

Without control of the later ratios, this is only a model for the tail. Do not present it as a theorem. For asymptotic series, terms eventually increase and the geometric argument ceases to apply; truncation near the least term is often the useful prescription.

Eigenvalue and variational calculations

For a normalized approximate eigenvector $|\psi\rangle$ in the domain of the self-adjoint Hamiltonian and a real energy estimate $E$, compute the full-space residual

$$|r\rangle = (H-E)|\psi\rangle, \qquad \eta_r = \|r\|.$$

The spectral theorem gives $\operatorname{dist}(E,\operatorname{spec}H)\le\eta_r$. To infer an eigenvector error, also separate the target eigenvalue from the rest of the spectrum: if every unwanted spectral value is at least $d$ from $E$, the norm outside the target subspace is at most $\eta_r/d$. A residual computed only inside a truncated matrix can vanish even when the omitted-space residual is large. Without gap information, it need not identify which eigenvector has been approximated. A variational upper bound on the ground-state energy is rigorous, but the difference between two trial energies is not by itself a rigorous error bar.

Transition probabilities and rates

First-order time-dependent perturbation theory requires the transition probability to remain small in the regime where depletion is neglected. A golden-rule rate additionally assumes an intermediate window

$$\tau_{\mathrm{corr}} \ll t \ll \Gamma^{-1},$$

where $\tau_{\mathrm{corr}}$ is a correlation or spectral-resolution time and $\Gamma^{-1}$ is the depletion timescale. The first inequality allows the finite-time line shape to resolve the continuum; the second keeps the initial state approximately undepleted.

If no such window exists, the formal rate may still be a useful coefficient, but it does not describe a sustained exponential process without additional resummation or open-system reasoning.

WKB and tunneling

Away from turning points, inspect $\hbar|p'|/p^2$ and the smoothness needed for higher derivatives; near a turning point use an appropriate matching solution. Define the dimensional barrier action $K=\int_{x_1}^{x_2}|p(x)|dx$ and the dimensionless ratio $B=K/\hbar$. For an opaque barrier, $T\propto e^{-2B}$. Holding the prefactor fixed,

$$\frac{T(B+\delta B)}{T(B)}=e^{-2\delta B}.$$

Even a small relative error in a large action can give a large probability error. Examine the action and prefactor separately; ground-state matching can change a prefactor by a constant even as the action grows.

Scattering amplitudes

For a Born expansion

$$f = f^{(1)} + f^{(2)} + \cdots,$$

the ratio $|f^{(2)}/f^{(1)}|$ is useful only where $f^{(1)}$ is not near zero. Near a diffraction minimum or symmetry-enforced node, compare absolute amplitudes against a characteristic scale or compare integrated observables.

Check channel completeness and flux normalization before using unitarity. Order-by-order perturbative unitarity can relate different orders, so the lowest Born amplitude need not satisfy the exact optical theorem by itself. Validity of the Born Approximation gives the method-specific criteria.

Example: A Small Gap Changes the Expansion

For $H=\begin{pmatrix}0&v\\v^*&\Delta\end{pmatrix}$ with $\Delta>0$, the lower shift begins as $-|v|^2/\Delta+|v|^4/\Delta^3+\cdots$. The ratio controlling the second term relative to the first is $(|v|/\Delta)^2$, not the dimensional number $|v|$. If the gap approaches zero, diagonalize the pair together. The two-level Stark example derives the exact eigenvalues and an exact relative-error formula.

Avoid False Precision

Printing more digits does not improve the physics. Numerical stability tests a chosen Hamiltonian; it does not test an interaction omitted from that Hamiltonian. Input uncertainty may also dominate the formal expansion error. Report these different limitations separately, and compare a representative range of parameters rather than relying on one point where errors might cancel.

Exercises

1. Estimate or bound?

A calculation includes terms through $O(\epsilon^2)$. The $O(\epsilon^2)$ correction is one percent of the $O(\epsilon)$ correction. No theorem about later coefficients is available. Is one percent a rigorous error bound?

Solution

No. The observed ratio is a useful truncation indicator, but later coefficients could grow or the first correction could be accidentally large or small. One may report that the computed term pattern suggests a percent-level correction to the leading perturbative contribution, while clearly labeling this as an estimate rather than a bound.

2. Compare the coupling with the gap

For the two-level Hamiltonian above, let $\Delta=2.0\,\mathrm{eV}$ and $|v|=0.10\,\mathrm{eV}$. Estimate the size of the first omitted correction relative to the leading energy shift.

Solution

The dimensionless parameter is

$$\epsilon = \frac{|v|}{\Delta} = 0.05.$$

The leading shift is $-|v|^2/\Delta$, and the next term is $|v|^4/\Delta^3$. Their magnitude ratio is

$$\frac{|v|^4/\Delta^3} {|v|^2/\Delta} = \epsilon^2 = 2.5\times10^{-3}.$$

The next term therefore suggests a correction of about $0.25\%$ of the leading shift. This remains an estimate unless the remainder is bounded independently.

3. Propagate an action error

A tunneling calculation gives $B=K/\hbar=5.0\pm0.2$ and uses $T\propto e^{-2B}$. What multiplicative uncertainty follows from the stated uncertainty in $B$?

Solution

At the endpoints,

$$\frac{T(B\pm0.2)}{T(B)} = e^{\mp0.4}.$$

Thus the upper factor is $e^{0.4}\approx1.49$ and the lower factor is $e^{-0.4}\approx0.67$. The linear estimate $\delta T/T\simeq-2\delta B=-2\delta K/\hbar$ gives about $40\%$, while the exponential propagation shows the asymmetric finite uncertainty more accurately.

4. Choose a scale at a node

The leading scattering amplitude vanishes at an angle $\theta_0$, while the next-order amplitude is nonzero. Why is $|f^{(2)}/f^{(1)}|$ a poor error measure there, and what should be reported instead?

Solution

The ratio divides by a quantity that tends to zero, so it diverges even when $f^{(2)}$ is small compared with a typical amplitude away from the node. Near $\theta_0$, the next nonzero term becomes the leading local prediction. Report its absolute size relative to a characteristic amplitude or cross-section scale, describe the angular width of the region where it matters, and compare with a higher-order or numerical result if available.

5. Separate two error budgets

A Rayleigh–Ritz energy changes by less than $10^{-8}$ when the basis is enlarged, but the Hamiltonian omits a relativistic correction expected at relative order $10^{-4}$. What accuracy is justified for the physical prediction?

Solution

The $10^{-8}$ stability diagnoses numerical basis truncation for the chosen nonrelativistic Hamiltonian. It does not test the omitted relativistic physics. Unless that correction is calculated or shown to cancel, the physical prediction is controlled only at roughly the $10^{-4}$ level. The two uncertainties should be reported separately.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, 1999.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315–397, 1972.
• T. Kato, Perturbation Theory for Linear Operators, corrected printing of the 2nd ed., Springer, 1995.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.