Error-Estimate Checklist

An approximation is not ready to report merely because the algebra is finished. A mature result identifies what was approximated, why the approximation should work, how large the neglected effects may be, and which observations would reveal failure.

This page is an operational companion to Small Parameters and Error Estimates. It is designed for the last pass through a calculation, a numerical notebook, or a draft manuscript. It does not replace a method-specific derivation.

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
$	R	\le B$ under stated assumptions
$R=O(\epsilon^{N+1})$ 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:

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

The Ten-Step Audit

Complete all ten steps before assigning a numerical error bar or a qualitative confidence statement.

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

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

and a ratio of interaction range to wavelength in a scattering problem. 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 limit is necessary evidence, but a single limit 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

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

Stability of $\Delta_N$ is evidence of numerical convergence, not automatically of physical-model accuracy. A converged calculation can solve the wrong Hamiltonian very precisely.

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.

A Compact Audit Record

The following record is short enough to place beside a calculation or in a notebook.

Field	What to Record
Target	Observable, kinematic point, and desired tolerance
Exact starting point	Hamiltonian, equation, or numerical model before approximation
Control parameters	Dimensionless ratios and the limits in which they are small or large
Singular regions	Degeneracies, thresholds, resonances, turning points, nodes, and endpoints
Retained content	Perturbative order, basis size, channels, operators, or saddle points kept
Omitted content	First neglected terms and effects outside the model
Internal checks	Units, normalization, symmetries, conservation laws, and residuals
External checks	Exact limits, independent numerics, or data
Error statement	Bound, asymptotic order, indicator, or observed discrepancy
Validity sentence	Plain-language domain in which the result is intended to apply

Method-Specific Diagnostics

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

Method	Primary Control	Useful Error Evidence	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 $|\psi\rangle$ with energy estimate $E$, compute the residual

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

A small residual shows that $|\psi\rangle$ is close to some spectral subspace when the relevant eigenvalue is isolated by a known gap. 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, evaluate the local diagnostic

$$\epsilon_{mathrm{WKB}}(x) = \left| \frac{\hbar p'(x)}{p(x)^2} \right|.$$

For a forbidden interval $[x_1,x_2]$, define

$$K = \frac{1}{\hbar} \int_{x_1}^{x_2} |p(x)|\,dx.$$

The leading transmission factor is proportional to $e^{-2K}$ when $K\gg1$ and the matching assumptions hold. An uncertainty $\delta K$ produces a multiplicative effect,

$$\frac{\delta T}{T} \simeq -2\,\delta K,$$

for $|\delta K|\ll1$. Thus a small relative error in the action can be important for an exponentially small probability. Check the exponent and prefactor separately.

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.

Worked Audit: A Nearly Degenerate Pair

Consider

$$H = \begin{pmatrix} 0 & v \\ v^* & \Delta \end{pmatrix}, \qquad \Delta>0.$$

The lower exact eigenvalue is

$$E_- = \frac{\Delta}{2} - \sqrt{ \frac{\Delta^2}{4} + |v|^2 }.$$

For $|v|/\Delta\ll1$,

$$E_- = -\frac{|v|^2}{\Delta} + \frac{|v|^4}{\Delta^3} + O\!\left( \frac{|v|^6}{\Delta^5} \right).$$

The audit reads as follows:

1. Target: the lower eigenvalue.
2. Control parameter: $\epsilon=|v|/\Delta$.
3. Denominator: the only unperturbed gap is $\Delta$.
4. Degeneracy: the expansion fails as $\Delta\to0$; the exact two-state diagonalization is then the correct leading problem.
5. Boundary and units: all matrix entries and eigenvalues have energy units; no boundary-condition issue arises in this finite model.
6. Symmetry: the eigenvalue depends on $|v|$, so a rephasing of either basis state cannot change it.
7. Limit: $E_-\to0$ as $v\to0$.
8. Benchmark: the exact eigenvalue is already available.
9. Omitted term: relative to the leading shift, the next correction has magnitude $\epsilon^2$.
10. Validity statement: the second-order shift $-|v|^2/\Delta$ is reliable for the lower level when $|v|\ll\Delta$; it is not uniform near the avoided crossing.

This example also shows why “small $v$” is incomplete. The comparison scale is the gap.

Worked Audit: A Golden-Rule Rate

Suppose a weak interaction couples an initial discrete state to continuum states and gives

$$\Gamma = \frac{2\pi}{\hbar} |V_{fi}|^2 \rho(E_f)$$

in a chosen normalization. A compact audit asks:

• Does the continuum vary slowly over the finite-time energy width $\hbar/t$?
• Is there a time window with $\tau_{\mathrm{corr}}\ll t\ll\Gamma^{-1}$?
• Are all symmetry-allowed channels included, with forbidden channels absent?
• Do the dimensions of $|V_{fi}|^2\rho(E_f)/\hbar$ give inverse time?
• Does a finite-time transition calculation approach a linear probability $P(t)\simeq\Gamma t$ in that window?
• Is the initial-state depletion still negligible?

The result should be reported as a weak-coupling, continuum-limit rate. It should not be extrapolated to arbitrarily long times using $P(t)=\Gamma t$, because a probability cannot grow without bound.

Worked Audit: A Born Cross Section

For a short-range potential, suppose the first Born approximation gives $f^{(1)}(\theta)$ and

$$\frac{d\sigma^{(1)}}{d\Omega} = |f^{(1)}(\theta)|^2.$$

Before using the result:

1. Check that $f^{(1)}$ has dimensions of length in the chosen normalization.
2. Form a dimensionless strength-and-range diagnostic appropriate to the potential and energy.
3. Compare with $f^{(2)}$ where feasible, or with numerically extracted phase shifts.
4. Increase the partial-wave cutoff until the benchmark cross section stabilizes.
5. Inspect threshold, forward-angle, and resonant regions separately; the approximation need not fail uniformly.
6. Test flux relations at the perturbative order at which they are expected to hold.

If $f^{(1)}$ has an angular zero, do not announce an infinite relative error because $f^{(2)}/f^{(1)}$ diverges. The first-order prediction is then locally suppressed, and the next nonzero amplitude sets the leading cross section near that angle.

Common Reporting Mistakes

• Precision without control: printing ten digits from a first-order approximation does not create ten-digit accuracy.
• A percentage without a scale: relative error near a zero can be arbitrarily large even when the absolute discrepancy is small.
• A next term called a bound: the next omitted term estimates a remainder only under additional assumptions about later terms.
• A single-point benchmark: agreement at one parameter value can conceal failure elsewhere.
• Dependent checks: two implementations of the same approximation are not independent physical validations.
• Numerical convergence confused with model accuracy: grid and basis convergence do not test omitted interactions.
• Input uncertainty ignored: uncertain masses, couplings, or potentials can dominate formal truncation error.
• Validity inferred from a good fit: adjustable parameters can absorb missing physics and hide a flawed approximation.

Exercises

1. Classify the evidence

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. Audit the two-level parameter

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 $K=5.0\pm0.2$ and uses $T\propto e^{-2K}$. What multiplicative uncertainty follows from the stated uncertainty in $K$?

Solution

At the endpoints,

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

Thus the upper factor is $e^{0.4}\approx1.49$ and the lower factor is $e^{-0.4}\approx0.67$. The linear estimate $\delta T/T\simeq-2\delta K$ 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.

Cross-Links

• Small Parameters and Error Estimates
• Common Failure Modes
• Approximation Decision Tree
• Method Comparison Table
• Formula Sheet
• Asymptotic Analysis
• Matrix Diagonalization
• Reproducibility Checklist audits the executable environment, validation, data, provenance, and licensing around a numerical result.

References

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