Small Parameters and Error Estimates

An approximation is trustworthy only when its error is controlled or at least diagnosed. The phrase “small perturbation” is incomplete until the relevant dimensionless ratio is stated.

This page gives the discipline used throughout the volume. For a compact final-pass audit, use the Error-Estimate Checklist.

Dimensionless Ratios

A quantity with units is not small by itself. A perturbing energy of $10^{-3}\,\mathrm{eV}$ may be tiny in atomic physics and huge in a nearly degenerate qubit. The meaningful quantity is a ratio such as

$$\epsilon_{mn} = \frac{\lvert V_{mn}\rvert} {\lvert E_m^{(0)}-E_n^{(0)}\rvert}.$$

If $\epsilon_{mn}\ll1$ for all states that significantly couple to the state of interest, nondegenerate perturbation theory may be controlled. If not, the method must be changed.

Energy Gaps and Denominators

Perturbation theory often fails because denominators become small. A typical first-order state correction contains terms of the form

$$\frac{V_{mn}}{E_n^{(0)}-E_m^{(0)}}.$$

Even if every matrix element of $V$ looks modest, the ratio can be large near degeneracy.

The practical rule is:

$$\text{small perturbation} \ne \text{small operator norm in isolation}.$$

It means small compared with the spectral separations relevant to the calculation.

Leading Order and Next Order

When an approximation gives an expansion

$$Q(\lambda) = Q^{(0)} + \lambda Q^{(1)} + \lambda^2Q^{(2)} + \cdots,$$

the leading-order result is only useful if the omitted terms are plausibly smaller:

$$\left| \lambda^2Q^{(2)} \right| \ll \left| \lambda Q^{(1)} \right|$$

or if there is another independent argument for truncation.

In practice one often estimates error by computing the next correction, comparing to a limiting case, or checking against a numerical benchmark.

Asymptotic Versus Convergent

Not every useful expansion converges. An asymptotic expansion may satisfy

$$Q(\lambda) - \sum_{r=0}^{N} \lambda^rQ^{(r)} = O(\lambda^{N+1})$$

as $\lambda\to0$ for fixed $N$, even though the infinite series diverges.

This means adding terms improves the answer only up to an optimal order. Beyond that, terms can grow. Many semiclassical and perturbative quantum expansions have this character.

For the mathematical definition of asymptotic expansions, stationary phase, and saddle-point estimates, see Asymptotic Analysis.

Variational Error

A variational estimate for the ground-state energy gives a bound:

$$E_0 \le \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}.$$

The error is one-sided for the energy but not necessarily for other observables. A trial state can give a good energy while giving a poor expectation value for a more sensitive operator.

Thus a variational calculation should report both the optimized energy and diagnostic checks on the trial state’s physical content.

WKB Error Diagnostics

For one-dimensional WKB, define

$$p(x)=\sqrt{2m(E-V(x))}.$$

A common validity condition away from turning points is that the local de Broglie wavelength changes slowly:

$$\left| \frac{d}{dx} \frac{\hbar}{p(x)} \right| \ll 1.$$

Equivalently, one often checks that the amplitude varies slowly compared with the phase. This condition necessarily fails near a simple turning point where $p(x)=0$, so connection formulas are not optional there.

Scattering Error Diagnostics

In scattering, a weak-potential expansion may be checked by comparing the scattered wave to the incident wave. Born approximation requires the scattered field generated by the potential to remain small enough that repeated scattering can be neglected.

Unitarity provides another diagnostic. If an approximate scattering amplitude violates probability conservation badly in a regime where unitarity should be visible, the approximation is being pushed too far.

Born Series gives the corresponding Neumann remainder bound when an appropriate operator norm is controlled. Validity of the Born Approximation separates that guarantee from practical range-and-strength estimates, accumulated-phase criteria, and observable-level numerical checks.

Error Bounds and Error Indicators

An error bound is a rigorous inequality. An error indicator is a diagnostic that suggests reliability but does not prove it.

Examples of error indicators include:

• size of the next perturbative correction,
• comparison with a solvable limiting case,
• symmetry checks,
• normalization checks,
• conservation-law checks,
• numerical diagonalization in a truncated basis,
• stability under changing a variational ansatz.

Good physics writing distinguishes these categories. Do not call an error indicator a bound unless it really is one.

Reporting Approximation Validity

Each method page in this volume should state:

• the control parameter,
• the leading retained order,
• the expected size of omitted terms,
• the known failure modes,
• at least one independent check when practical.

A result without a validity statement is not yet a mature approximation result.

Common Mistakes

• Saying “small” without a dimensionless ratio.
• Ignoring near-degenerate denominators.
• Treating an asymptotic expansion as a convergent series.
• Reporting a variational energy without checking the wavefunction.
• Applying WKB at turning points without matching.
• Trusting a scattering approximation after it visibly violates unitarity.

Cross-Links

• Volume Overview
• Approximation Map
• Choosing a Method
• Perturbative, Variational, and Asymptotic Thinking
• Common Failure Modes
• Error-Estimate Checklist
• Rabi Formula in the Weak-Drive Limit gives an explicit fixed-time expansion that fails under long-time scaling.
• Asymptotic Analysis
• Matrix Diagonalization
• Quantum Harmonic Oscillator

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
• 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.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315-397, 1972.

Exercises

1. A first-order correction is $0.02$ in chosen units and a second-order correction is $0.015$. What does this suggest about the expansion?

Solution

The second-order correction is not much smaller than the first-order correction. This suggests the expansion is not well controlled at the chosen parameter value, or that the first correction is accidentally small. One should check higher orders, use a different method, or compare with a numerical benchmark.

2. Explain why a variational upper bound on energy does not automatically imply accurate expectation values for all observables.

Solution

The variational principle bounds the ground-state energy expectation. A trial state can have an energy close to the true ground-state energy while still having incorrect short-distance behavior, long-distance tails, or correlations. Observables sensitive to those features can be inaccurate even when the energy looks good.