Common Failure Modes

An approximation fails when one of its supporting assumptions stops controlling the omitted physics. The warning may be dramatic, such as a probability larger than one, or deceptively mild, such as a stable-looking energy whose wavefunction has the wrong tail.

This page is a diagnostic, not a catalog of embarrassing algebra. Each failure mode is organized around four questions:

1. What symptom appears?
2. Which assumption was violated?
3. What quantitative test exposes the problem?
4. Which reorganization repairs it?

The first response to a failed approximation should be structural. Adding more terms to the same expansion is useful only if the expansion remains the right one.

Rapid Triage

Symptom	Likely failure	Immediate test	Typical repair
A parameter is called small but carries units	No dimensionless control parameter	Nondimensionalize with physical scales	Rewrite in scaled variables
State corrections or mixing angles are large	Near degeneracy or small denominator	Compare coupling with spectral gap	Diagonalize an enlarged subspace
Levels cross in a truncated calculation	Off-diagonal mixing omitted	Inspect symmetry-allowed couplings	Use a two-level or degenerate treatment
A rate changes with the observation window	Golden-rule regime absent	Compare correlation, depletion, and recurrence times	Use finite-time dynamics or a reduced coherent model
WKB amplitude diverges	Turning point or caustic	Evaluate $\hbar\lvert p'\rvert/\lvert p\rvert^2$	Use Airy matching or a uniform approximation
Higher orders first improve and then worsen	Divergent asymptotic series	Track term ratios and the least term	Truncate optimally and estimate the remainder
Born cross section is too large or violates partial-wave scales	Repeated scattering or resonance	Extract phase shifts or test unitarity	Solve the integral equation or use partial waves
Forward Coulomb integral diverges	Short-range asymptotics used for a long-range force	Check the large-$r$ tail and detector acceptance	Use Coulomb-distorted states or screening
A corrected state has norm different from one	Intermediate normalization mistaken for unit normalization	Expand $\langle\psi\vert\psi\rangle$ consistently	Add the parallel normalization correction
Variational energy is excellent but observables drift	Trial state error hidden by energy stationarity	Compute variance, residuals, and sensitive observables	Enlarge the trial family
Approximate scattering loses probability	Unitarity checked inconsistently or approximation pushed too far	Compare both sides of the optical theorem at the same order	Include the required order or use a unitary method
A leading term unexpectedly vanishes	Exact or approximate symmetry	Evaluate transformation properties and selection rules	Reorganize around the first allowed contribution

The table identifies starting points. The detailed diagnosis still depends on the model, observable, normalization convention, and intended regime.

Failure 1: Expanding in a Dimensional Quantity

Symptom.

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.

Mechanism.

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 $\hbar$.

For the quartic oscillator

$$H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2 + g x^4,$$

the coupling $g$ has units. Using the oscillator length

$$a=\sqrt{\frac{\hbar}{m\omega}},$$

the dimensionless ratio is

$$\epsilon = \frac{g a^4}{\hbar\omega} = \frac{g\hbar}{m^2\omega^3}.$$

The statement $\epsilon\ll1$ is meaningful; the statement $g\ll1$ is not.

Diagnostic.

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

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

Repair.

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.

Failure 2: Ignoring Near Degeneracy

Symptom.

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.

Mechanism.

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,

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

Diagnostic.

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.

Repair.

Diagonalize $PVP$ inside the degenerate or quasi-degenerate subspace $P\mathcal H$. 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.

Failure 3: Trusting First Order Near a Level Crossing

Symptom.

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.

Mechanism.

The local crossing model

$$H(s) = \begin{pmatrix} \alpha s & g\\ g^* & -\alpha s \end{pmatrix}$$

has exact eigenvalues

$$E_\pm(s) = \pm \sqrt{\alpha^2s^2+\lvert g\rvert^2}.$$

The uncoupled lines $\pm\alpha s$ cross at $s=0$, but any nonzero symmetry-allowed $g$ produces a minimum gap $2\lvert g\rvert$. An expansion valid for $\lvert\alpha s\rvert\gg\lvert g\rvert$ cannot be continued uniformly through the crossing.

Diagnostic.

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.

Repair.

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.

Failure 4: Using Fermi’s Golden Rule Outside Its Time Window

Symptom.

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.

Mechanism.

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.

Diagnostic.

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.

Repair.

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 canonical regime statement is on Fermi’s Golden Rule.

Failure 5: Applying WKB at a Turning Point

Symptom.

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.

Mechanism.

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.

Diagnostic.

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.

Repair.

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.

Failure 6: Treating an Asymptotic Series as Convergent

Symptom.

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.

Mechanism.

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.

Diagnostic.

• 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.

Repair.

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.

Failure 7: Using the Born Approximation for Strong or Resonant Scattering

Symptom.

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.

Mechanism.

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.

Diagnostic.

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.

Repair.

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.

Failure 8: Forgetting Long-Range Coulomb Structure

Symptom.

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.

Mechanism.

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.

Diagnostic.

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.

Repair.

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.

Failure 9: Losing Normalization in State Corrections

Symptom.

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.

Mechanism.

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:

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

Diagnostic.

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.

Repair.

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.

Failure 10: Mistaking a Good Variational Energy for a Good Wavefunction

Symptom.

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

Mechanism.

For

$$\lvert\psi\rangle = \sqrt{1-\delta^2}\,\lvert0\rangle + \delta\lvert\chi\rangle, \qquad \langle0\vert\chi\rangle=0,$$

the energy error is

$$\mathcal R[\psi]-E_0 = \delta^2 \left( \langle\chi\rvert H\lvert\chi\rangle-E_0 \right).$$

The energy is stationary at the exact state, so its leading error is quadratic in the state contamination. A general observable can have a linear error through $\langle0\vert A\lvert\chi\rangle$.

Diagnostic.

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 useful evidence, not a universal state-distance bound without spectral information.

Repair.

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.

Failure 11: Checking Unitarity at the Wrong Order

Symptom.

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.

Mechanism.

For short-range elastic scattering,

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

For a real potential, the first Born amplitude is often real and of order $V$. 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.

Diagnostic.

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.

Repair.

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.

Failure 12: Ignoring Symmetry-Enforced Zeros

Symptom.

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.

Mechanism.

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.

Diagnostic.

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.

Repair.

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.

The systematic treatment belongs to 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.

Recovery Workflow

When a calculation fails a diagnostic:

1. Freeze the claim. Record the model, observable, parameter values, normalization, and retained order.
2. Identify the singular structure. Look for a small gap, resonance, threshold, turning point, long-range tail, secular time, or omitted symmetry sector.
3. Choose a local replacement. Use an enlarged subspace, uniform approximation, finite-time kernel, pole expansion, or distorted asymptotic state.
4. Match to the original regime. The replacement should reproduce the old result where the old control parameter becomes small again.
5. Add an independent benchmark. Use an exact limit, inequality, conservation law, or converged numerical result.
6. Restate the domain. Report where the repaired approximation is intended to work and which failure is expected next.

Failure analysis is successful when it explains both why the old method broke and why the replacement should work.

Exercises

Near-degenerate mixing

For the two-level Hamiltonian in Failure 2, 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 potential, $f^{(1)}$ 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.

Cross-Links

• Volume Overview
• Small Parameters and Error Estimates
• Perturbative, Variational, and Asymptotic Thinking
• Notation and Conventions
• Approximation Decision Tree
• Error-Estimate Checklist
• Nondegenerate Perturbation Theory
• Fermi’s Golden Rule
• Turning Points and Connection Formulas
• First Born Approximation
• Coulomb Scattering
• Variational Principle
• Optical Theorem
• Selection Rules

References

• 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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vols. 1–2, Wiley, 1977.
• T. Kato, Perturbation Theory for Linear Operators, corrected printing of the 2nd ed., Springer, 1995.
• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
• F. W. J. Olver and R. Wong, DLMF Chapter 2: Asymptotic Approximations, NIST Digital Library of Mathematical Functions.
• 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.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978.