Common Failure Modes

An approximation can fail dramatically, by predicting a probability greater than one, or quietly, by giving a stable energy but an inaccurate wavefunction. The useful question is which assumption failed and what simpler description should replace it. This guide connects recognizable symptoms to quantitative checks and remedies.

Helpful background. Error estimates relates accuracy to the target observable, and the comparison of approximation principles distinguishes coefficients, bounds, and asymptotic remainders.

Expanding in a Dimensional Quantity

A derivation says that $g$, $V$, a field strength, or a frequency is small without stating what it is small compared with. Changing units then changes the numerical size of the alleged expansion parameter.

Only dimensionless ratios can order a physical expansion. A perturbing energy must be compared with a gap, a length with a wavelength, a time with a dynamical time, and an action with $\hbar$.

The quartic-oscillator example shows how to obtain a dimensionless coupling by dividing the Hamiltonian by its natural energy scale.

Check.

Nondimensionalize the Hamiltonian and observable before expanding. Every retained and omitted term should be assigned an order in dimensionless parameters. If several independent scales exist, a single global parameter may be insufficient.

For state mixing, a more local ratio is often

$$\epsilon_{mn} = \frac{\lvert V_{mn}\rvert} {\lvert E_m-E_n\rvert}.$$

What to try.

Choose characteristic units from the physical problem, not from convenience alone. Report the parameter range in which all relevant ratios are small. If two ratios compete, use a multiscale expansion and state the assumed ordering between them.

Ignoring Near Degeneracy

Nondegenerate state corrections become large, second-order energy shifts diverge as a gap closes, or the result depends sensitively on an arbitrary basis inside a nearly degenerate sector.

Consider

$$H = \begin{pmatrix} E_0+\Delta/2 & \lambda v\\ \lambda v^* & E_0-\Delta/2 \end{pmatrix}.$$

The exact eigenvalues are

$$E_\pm = E_0 \pm \sqrt{ \left(\frac{\Delta}{2}\right)^2 + \lambda^2\lvert v\rvert^2 }.$$

Nondegenerate perturbation theory expands in

$$\epsilon_{\mathrm{mix}} = \frac{\lvert\lambda v\rvert}{\lvert\Delta\rvert}.$$

When this ratio is not small, the unperturbed basis states are not close to the exact eigenstates. The mixing angle obeys, up to a phase convention,

$$\tan(2\theta) = \frac{2\lvert\lambda v\rvert}{\Delta}.$$

Check.

Construct the coupling matrix among all states inside the energy window set by the perturbation. Compare its off-diagonal scale with both internal splittings and the gap to states outside the proposed subspace.

What to try.

For exact degeneracy, diagonalize $PVP$ within $P\mathcal H$. For near degeneracy, keep the small splittings in $PH_0P$ as well and diagonalize $P(H_0+V)P$, adding corrections from the complementary space at the required order. If outside states also have large coupling-to-gap ratios, enlarge $P$. Only after the adapted zeroth-order states are found should ordinary corrections from the complementary space be added.

See Degenerate Perturbation Theory.

Trusting First Order Near a Level Crossing

Two approximate energy curves cross, but the underlying states have the same symmetry and a nonzero matrix element between them. Observables derived from the uncoupled states jump discontinuously at the crossing.

In a two-state model with diagonal entries $\alpha s$ and $-\alpha s$ and off-diagonal entries $g$ and $g^*$, the eigenvalues are $\pm\sqrt{\alpha^2s^2+\lvert g\rvert^2}$. Any nonzero $g$ opens a minimum gap $2\lvert g\rvert$. An expansion valid far from the crossing cannot be continued uniformly through it.

Check.

Ask whether the two states carry different exact symmetry quantum numbers. If they do, the off-diagonal coupling may vanish and a true crossing can remain. If they have the same symmetry, compute the coupling rather than assuming it is zero.

Track the ratio

$$\frac{\lvert g\rvert}{\lvert\alpha s\rvert}.$$

It diverges in the crossing region.

What to try.

Use a two-state effective Hamiltonian or degenerate perturbation theory near the crossing and match it to the outer nondegenerate expansions. If $s$ varies in time, the dynamical transition problem belongs to Landau–Zener Transition.

Using Fermi’s Golden Rule Outside Its Time Window

A supposed constant rate depends strongly on observation time, a discrete final state undergoes coherent oscillations, the initial population is substantially depleted, or finite-system recurrences become visible.

For a time-independent perturbation switched on for duration $T$, first-order transition probability between two discrete levels contains

$$P_{i\to f}^{(1)}(T) = \frac{4\lvert V_{fi}\rvert^2}{\hbar^2} \frac{ \sin^2(\omega_{fi}T/2) }{ \omega_{fi}^2 }.$$

At exact resonance this grows as $T^2$, not linearly. Linear growth emerges after summing over a sufficiently dense continuum whose phases dephase across an energy width of order $\hbar/T$.

The golden-rule window is schematically

$$\tau_{\mathrm{corr}} \ll T \ll \min\!\left( \Gamma^{-1}, t_{\mathrm{rec}} \right),$$

where $\tau_{\mathrm{corr}}$ is a microscopic correlation or dephasing time, $\Gamma^{-1}$ is the depletion time, and $t_{\mathrm{rec}}$ is a recurrence time.

Check.

Check:

• whether the final spectrum is effectively continuous on the resolution scale $\hbar/T$;
• whether $\rho(E)\lvert V(E)\rvert^2$ is smooth across that window;
• whether the initial population remains near one;
• whether a coherent few-level description gives oscillations instead of irreversible loss;
• whether the system size makes recurrences relevant.

What to try.

Use the finite-time transition amplitude when the energy delta function is unresolved. Use Rabi or reduced few-level dynamics for coherent transitions. Include depletion through a controlled resummation or open-system description when the Markov assumptions are justified.

The time-window assumptions are developed in Fermi’s Golden Rule.

Applying WKB at a Turning Point

The WKB amplitude $1/\sqrt{p(x)}$ diverges even though the exact wavefunction remains finite, or an independently chosen allowed-region and forbidden-region solution cannot be matched consistently.

For

$$p(x)=\sqrt{2m\bigl(E-V(x)\bigr)},$$

the local WKB diagnostic is

$$\eta_{\mathrm{WKB}}(x) = \frac{\hbar\lvert p'(x)\rvert} {\lvert p(x)\rvert^2}.$$

At a simple turning point $x_t$ with $E=V(x_t)$ and $V'(x_t)\ne0$,

$$\eta_{\mathrm{WKB}}(x) \propto \lvert x-x_t\rvert^{-3/2},$$

so the approximation necessarily fails in a neighborhood of $x_t$.

Linearizing the potential identifies the turning-point length

$$\ell_t = \left( \frac{\hbar^2} {2m\lvert V'(x_t)\rvert} \right)^{1/3}.$$

Inside this region the local equation reduces to the Airy equation.

Check.

Plot or estimate $\eta_{\mathrm{WKB}}(x)$, locate every zero of $p(x)$, and determine whether turning points are isolated and simple. Coalescing turning points require a different uniform model than a single Airy function.

What to try.

Use the local Airy solution and Turning Points and Connection Formulas. For abrupt finite jumps, exact piecewise matching is usually more natural than WKB. For caustics or coalescing saddles, use an appropriate uniform semiclassical approximation.

Treating an Asymptotic Series as Convergent

Successive orders improve the answer at first and then make it worse, coefficients grow factorially, or the numerical sum depends violently on how many terms are retained.

For formal terms

$$T_n(\epsilon)=a_n\epsilon^n,$$

track

$$r_n(\epsilon) = \left\lvert \frac{T_{n+1}(\epsilon)} {T_n(\epsilon)} \right\rvert.$$

If $a_n$ grows like $n!A^{-n}$, then $r_n\sim n\epsilon/A$. For fixed $\epsilon$, terms decrease only until $n$ is of order $A/\epsilon$ and then increase. The infinite series diverges even though a finite partial sum can be highly accurate.

Check.

• Inspect large-order coefficient ratios.
• Compare several consecutive truncations rather than only the highest available order.
• Locate the least term.
• Distinguish an error bound from the heuristic size of the next term.
• Check whether the expansion is uniform in position, time, or another parameter.

What to try.

Truncate near the least term and report the truncation prescription. When justified, use Borel or another resummation method and state the analytic assumptions. Include exponentially small sectors when the desired precision reaches beyond all algebraic orders.

Perturbative, Variational, and Asymptotic Thinking gives a factorially divergent integral with an explicit remainder bound.

Using the Born Approximation for Strong or Resonant Scattering

The Born cross section approaches or exceeds partial-wave unitarity scales, successive Born terms are not suppressed, or a weak change in potential strength produces a large low-energy response.

The Lippmann–Schwinger equation is

$$\lvert\psi_{\mathbf k}^{(+)}\rangle = \lvert\mathbf k\rangle + G_0^{(+)}(E)V \lvert\psi_{\mathbf k}^{(+)}\rangle.$$

The first Born approximation replaces the exact state on the right by the incident state. It neglects repeated scattering. That replacement is uncontrolled when $G_0^{(+)}V$ acts strongly on the relevant wave, even if the potential looks modest pointwise.

For elastic central scattering,

$$\sigma_\ell = \frac{4\pi}{k^2} (2\ell+1) \sin^2\delta_\ell \le \frac{4\pi}{k^2} (2\ell+1).$$

A shallow bound or virtual state near threshold can make the scattering length much larger than the potential range. This is a nonperturbative low-energy response.

Check.

Compare Born predictions with:

• the size of a second Born contribution;
• numerical or analytic phase shifts;
• partial-wave unitarity bounds;
• the scattering length and effective range;
• the proximity of bound-state or resonance poles.

What to try.

Solve the radial equation or Lippmann–Schwinger equation nonperturbatively, use partial waves, or match a low-energy effective-range description. Near a resonance, organize the pole contribution at leading order rather than treating it as a small correction.

See Validity of the Born Approximation for operational tests, First Born Approximation for the leading formula, and Bound States and Scattering Poles for the pole mechanism.

Forgetting Long-Range Coulomb Structure

The standard plane-wave-plus-outgoing-spherical-wave asymptotic form is inconsistent, partial-wave sums converge poorly, the total cross section diverges in the forward direction, or a short-range scattering length is assigned to an unscreened $1/r$ potential.

A Coulomb potential decays too slowly for ordinary short-range asymptotic states. The exact waves acquire logarithmic long-range phases, and the partial waves acquire Coulomb phases

$$\sigma_\ell = \arg\Gamma(\ell+1+i\eta),$$

with the sign of the Sommerfeld parameter $\eta$ tied to the interaction convention.

The Rutherford differential cross section behaves as

$$\frac{d\sigma}{d\Omega} \propto \frac{1}{\sin^4(\theta/2)}.$$

Its integral diverges as $\theta\to0$ for an ideal unscreened interaction and perfect angular acceptance.

Check.

Inspect the potential tail before choosing boundary conditions. State whether the interaction is screened, whether a forward cone is excluded, and which Coulomb phase convention is used. Do not infer convergence from agreement of the Born magnitude with Rutherford scattering; the phase structure remains nonperturbative in the ordinary short-range setup.

What to try.

Use exact or Coulomb-distorted asymptotic states. For a physically screened interaction, compute with the screening scale and take only controlled limits of finite observables. Report beam size, detector acceptance, or plasma screening when they regulate the forward region.

See Coulomb Scattering.

Losing Normalization in State Corrections

A corrected state has norm $1+O(\lambda^2)$ rather than one, expectation values depend on an unnoticed convention, or a component parallel to the reference state appears to be missing.

Intermediate normalization commonly imposes

$$\langle n^{(0)}\vert n(\lambda)\rangle=1.$$

Then

$$\lvert n(\lambda)\rangle_{\mathrm{int}} = \lvert n^{(0)}\rangle + \lambda\lvert n^{(1)}\rangle + \lambda^2\lvert n^{(2)}_\perp\rangle +\cdots,$$

with corrections chosen orthogonal to $\lvert n^{(0)}\rangle$. This state is convenient but not unit normalized:

$$\langle n(\lambda)\vert n(\lambda)\rangle_{\mathrm{int}} = 1 + \lambda^2 \langle n^{(1)}\vert n^{(1)}\rangle + O(\lambda^3).$$

The normalized state through second order contains a parallel correction:

$$\lvert n(\lambda)\rangle_{\mathrm{norm}} ={} \left[ 1 - \frac{\lambda^2}{2} \langle n^{(1)}\vert n^{(1)}\rangle \right] \lvert n^{(0)}\rangle + \lambda\lvert n^{(1)}\rangle + \lambda^2\lvert n^{(2)}_\perp\rangle + O(\lambda^3).$$

Check.

Expand the norm and every expectation-value denominator through the same order as the numerator. State whether the calculation uses intermediate normalization, unit normalization, or another phase and gauge convention.

What to try.

Apply the normalization factor consistently or keep the Rayleigh quotient denominator explicit. Do not mix state corrections derived in one convention with formulas assuming another. Remember that a state vector’s overall phase remains conventional even after its norm is fixed.

Mistaking a Good Variational Energy for a Good Wavefunction

The optimized energy is stable while densities, tails, contact values, correlation functions, or transition matrix elements change substantially when the ansatz is enlarged.

The variational comparison shows why this happens: the energy error is quadratic in a small admixture orthogonal to the exact eigenstate, while a general observable can change linearly.

Check.

In addition to the energy, compute:

• the energy variance

  $$\sigma_H^2 = \langle H^2\rangle-\langle H\rangle^2;$$

• the residual norm $\lVert(H-E)\lvert\psi\rangle\rVert$;

• observables sensitive to short- and long-distance behavior;

• stability under nested trial spaces;

• exact virial, cusp, symmetry, or sum-rule constraints when available.

Zero energy variance means the state lies in a single eigenspace. A small variance is a useful check, but does not give a universal bound on distance from a specified eigenstate without spectral information. In particular, zero variance does not identify the ground state: every exact excited eigenstate also has zero variance.

What to try.

Enlarge the ansatz according to the missing physics rather than merely adding parameters. Enforce exact symmetries and boundary behavior, add correlation structure, and validate the observables that the calculation is meant to predict.

See Variational Principle and Trial Wavefunctions. The dedicated Common Variational Pitfalls guide separates this state-quality failure from domain, symmetry, optimization, and estimator failures.

Checking Unitarity at the Wrong Order

An approximate scattering amplitude violates the optical theorem, produces $\lvert S_\ell\rvert\ne1$ in a nominally elastic problem, or appears to create or destroy flux.

For short-range elastic scattering,

$$\sigma_{\mathrm{tot}} = \frac{4\pi}{k} \operatorname{Im}f(0).$$

For a real local potential, the first Born forward amplitude is real and of order $V$. At other angles a real potential without inversion symmetry can have a complex Fourier transform. Its squared magnitude gives a cross section of order $V^2$, while the imaginary forward contribution required by the optical theorem first appears in the second Born amplitude, also at order $V^2$.

Comparing $\operatorname{Im}f^{(1)}(0)$ with $\lvert f^{(1)}\rvert^2$ mixes perturbative orders. It diagnoses bookkeeping, not necessarily a failed weak-potential expansion.

Check.

Expand both sides of every unitarity relation to the same order. In partial waves, inspect

$$S_\ell=e^{2i\delta_\ell}$$

for a single elastic channel, or $\lvert S_\ell\rvert\le1$ for an elastic element when inelastic channels are open. Include all open channels in the probability balance.

What to try.

Include the Born order required by the unitarity relation, or use a phase-shift, $K$-matrix, or direct numerical method that preserves unitarity in its construction. Do not hide a large violation with an ad hoc normalization factor; determine whether omitted repeated scattering is genuinely small.

See Born Series for the second-order shell term, Optical Theorem for the exact relation, and Unitarity Bridge for the QFT translation.

Ignoring Symmetry-Enforced Zeros

A leading correction or transition amplitude vanishes unexpectedly, a numerical result produces a tiny value sensitive to roundoff, or a calculation interprets zero first order as proof that the full effect is absent.

Let a parity operator act as

$$\mathcal P\lvert i\rangle = \pi_i\lvert i\rangle, \qquad \mathcal P\lvert f\rangle = \pi_f\lvert f\rangle,$$

and let

$$\mathcal P O\mathcal P^{-1} = \pi_O O.$$

Then

$$\langle f\vert O\lvert i\rangle = \pi_f\pi_O\pi_i \langle f\vert O\lvert i\rangle.$$

If $\pi_f\pi_O\pi_i=-1$, the matrix element vanishes exactly while the symmetry assumptions hold.

Check.

Before computing, classify states and operators under every exact symmetry: parity, rotations, translations, time reversal where applicable, particle exchange, and internal symmetries. Distinguish:

• an exact zero protected by the full Hamiltonian;
• a zero at one perturbative order;
• a zero lifted by a symmetry-breaking parameter;
• an accidental cancellation with no protection.

What to try.

Choose symmetry-adapted states and block-diagonalize before approximating. If the leading term is forbidden, identify the first allowed operator or perturbative order and revise the error estimate accordingly. If the environment or perturbation breaks the symmetry, include the breaking parameter explicitly.

For the systematic treatment, see Selection Rules and Symmetry Constraints on Hamiltonians.

False Successes

Some approximations fail without displaying a large number or divergence.

Accidental cancellation

A small leading correction may result from cancellation rather than a small expansion parameter. Compare the next several terms and inspect symmetry. If $Q^{(1)}$ is anomalously small, the ratio $Q^{(2)}/Q^{(1)}$ can be misleading in either direction.

Convergence inside the wrong model

A basis calculation can converge perfectly for a truncated Hamiltonian with the wrong boundary condition, omitted channel, or incorrect long-range tail. Numerical convergence controls discretization or truncation error, not model error.

Agreement of nonindependent methods

Two calculations may agree because they share the same approximation, convention mistake, or fitted input. Prefer checks with different failure mechanisms: a variational bound against direct diagonalization, WKB against exact high-lying levels, or Born scattering against phase-shift unitarity.

Fitting away the diagnostic

An adjustable parameter can reproduce one observable while concealing incorrect scaling elsewhere. Validate predictions not used in the fit and vary the regime, not merely the fit quality.

Return to a regime where both methods work

After changing the approximation, check that the new description reproduces the old result where the old assumptions become valid again. An enlarged two-state treatment should reduce to ordinary perturbation theory far from the avoided crossing. A turning-point solution should match WKB away from the turning region. This overlap is a useful check on both the repair and its normalization.

Exercises

Near-degenerate mixing

For the two-level Hamiltonian in the near-degeneracy section, take $\Delta=0.02\,\mathrm{eV}$ and $\lvert\lambda v\rvert=0.01\,\mathrm{eV}$. Evaluate the mixing diagnostic and decide whether ordinary nondegenerate perturbation theory is reliable.

Solution

The ratio is

$$\epsilon_{\mathrm{mix}} = \frac{0.01}{0.02} = 0.5.$$

This is not much smaller than one. The mixing angle satisfies $\tan(2\theta)=1$, so $\theta=\pi/8$ up to conventions, which is substantial. The coupled two-state subspace should be diagonalized rather than treated by ordinary nondegenerate state corrections.

Normalize a corrected state

Suppose

$$\lvert\psi\rangle_{\mathrm{int}} = \lvert0\rangle + \lambda\lvert1\rangle, \qquad \langle0\vert1\rangle=0, \qquad \langle1\vert1\rangle=c.$$

Find the factor that normalizes this state through order $\lambda^2$.

Solution

The norm is

$$\langle\psi\vert\psi\rangle_{\mathrm{int}} = 1+\lambda^2c.$$

Therefore

$$\frac{1}{\sqrt{1+\lambda^2c}} = 1-\frac{\lambda^2c}{2} +O(\lambda^4).$$

The normalized state through the required order is

$$\lvert\psi\rangle_{\mathrm{norm}} = \left(1-\frac{\lambda^2c}{2}\right) \lvert0\rangle + \lambda\lvert1\rangle + O(\lambda^3),$$

where additional genuine second-order components would be included separately.

Golden-rule window

A finite system has $\tau_{\mathrm{corr}}=10^{-13}\,\mathrm{s}$, $\Gamma^{-1}=10^{-8}\,\mathrm{s}$, and $t_{\mathrm{rec}}=10^{-10}\,\mathrm{s}$. Give a plausible golden-rule observation window and identify the limiting late-time scale.

Solution

The required ordering is

$$10^{-13}\,\mathrm{s} \ll T \ll \min(10^{-8},10^{-10})\,\mathrm{s}.$$

Thus a window such as $10^{-12}\,\mathrm{s}\lesssim T\lesssim10^{-11}\,\mathrm{s}$ may be plausible, subject to smooth spectral data. Recurrence at $10^{-10}\,\mathrm{s}$ limits the late-time side before substantial depletion does.

WKB turning region

For a simple turning point with slope $\lvert V'(x_t)\rvert=F$, derive the only length scale built from $\hbar$, $m$, and $F$ and explain its meaning.

Solution

Balancing the kinetic scale $\hbar^2/(2m\ell^2)$ with the linear-potential scale $F\ell$ gives

$$\frac{\hbar^2}{2m\ell^2} \sim F\ell.$$

Hence

$$\ell = \left( \frac{\hbar^2}{2mF} \right)^{1/3}.$$

This is the turning-point region in which the linearized Schrödinger equation has Airy solutions and ordinary WKB is not uniform.

Order-consistent unitarity

For a real weak local potential, $f^{(1)}(0)$ is real and of order $V$. Explain which terms must be compared in the optical theorem through order $V^2$.

Solution

The total cross section computed from the leading amplitude is

$$\sigma_{\mathrm{tot}}^{(2)} \sim \int d\Omega\, \lvert f^{(1)}(\Omega)\rvert^2,$$

which is order $V^2$. It must be compared with

$$\frac{4\pi}{k} \operatorname{Im}f^{(2)}(0),$$

also order $V^2$, not with $\operatorname{Im}f^{(1)}(0)=0$. The second Born amplitude supplies the imaginary part required by perturbative unitarity.

Symmetry changes the leading order

An odd perturbation $V$ acts on a nondegenerate even-parity state. What is the first-order energy correction, and what does its value imply about the leading error scale?

Solution

The first-order correction is

$$E^{(1)} = \langle n\vert V\lvert n\rangle.$$

The integrand or matrix element is odd under parity, so $E^{(1)}=0$. This does not mean the energy is unchanged. The leading correction is generally second order in the perturbation, so an error estimate based on a generic nonzero first-order term must be reorganized.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, 1999. Secular terms, nonuniform expansions, and optimal truncation.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315–397, 1972. Turning points and uniform approximations.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978. Spectral assumptions behind variational and residual estimates.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020. Degenerate perturbation theory, transitions, and selection rules.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006. Scattering boundary conditions and unitarity.