Approximation and Scattering Formulas

Approximation formulas are useful only together with their controlling assumptions. This index routes a calculation to the appropriate formula card and identifies the branch points that change the answer: degeneracy, a continuum limit, boundary phases, open scattering channels, long-range interactions, or a breakdown of weak coupling.

The cards are a lookup layer. Their canonical links lead to the pages that own the derivations, interpretations, and broader examples.

Choose by task

Question	Start here	First decision
How does an isolated level shift?	First-Order Perturbation Theory	Is the level nondegenerate and separated from nearby levels?
What is the leading shift when the diagonal first-order term vanishes?	Second-Order Perturbation Theory	Are the denominators safely larger than the couplings?
What transition rate does a weak drive produce?	Fermi Golden Rule	Is there a dense final spectrum and a valid long-time window?
Can a trial state bound the ground energy?	Variational Bound	Is the state in the Hamiltonian or quadratic-form domain?
Can a smooth classical orbit estimate bound levels?	WKB Quantization	What phase belongs to each turning point or boundary?
How does an amplitude become an observable area?	Scattering Cross Section	Are outgoing speeds, spins, and final-state counting included?
Can a weak potential be Fourier transformed into an amplitude?	Born Approximation	Is repeated scattering small in the interaction region?
Does the forward amplitude satisfy inclusive flux conservation?	Optical Theorem	Are all open channels included in the total?

Card catalog

Stationary spectrum corrections

First-Order Perturbation Theory starts from

$$H(\lambda)=H_0+\lambda V$$

and, for an isolated nondegenerate eigenstate, gives

$$E_n^{(1)} = \langle n^{(0)}\rvert V \lvert n^{(0)}\rangle.$$

The card also gives the first state and observable corrections, the near-degeneracy diagnostic, and the required degenerate branch “diagonalize $PVP$ first.”

Second-Order Perturbation Theory collects

$$E_n^{(2)} = \sum_{m\ne n} \frac{ \lvert V_{mn}\rvert^2 }{ E_n^{(0)}-E_m^{(0)} }$$

along with explicit $\lambda^2W$ terms, reduced-resolvent and Dalgarno–Lewis forms, sign checks for the ground state, continuum contributions, and second-order effective matrices for a degenerate subspace.

Use these cards when a solved $H_0$ is genuinely close to $H$ in the spectral sector of interest. A small diagonal matrix element does not control off-diagonal mixing.

Transition rates

Fermi Golden Rule gives the state-sum form

$$\Gamma_{i\to f} = \frac{2\pi}{\hbar} \lvert V_{fi}\rvert^2 \delta(E_f-E_i)$$

and its density-of-states version

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

The card begins from the finite-time sinc-squared transition probability, then states the long-time, weak-depletion, many-final-level window needed to replace that profile by an energy delta distribution. It also separates absorption and emission conventions, channel sums, branching fractions, initial mixtures, lifetimes, and linewidth caveats.

Use the finite-time probability rather than a constant rate when only one or a few final levels lie inside the energy-resolution window.

Bounds and semiclassical spectra

Variational Bound uses the Rayleigh quotient

$$\mathcal R[\psi] = \frac{ \langle\psi\rvert H\lvert\psi\rangle }{ \langle\psi\vert\psi\rangle } \ge E_0.$$

It supplies a rigorous upper bound when the trial state is admissible. The card also covers parameter stationarity, residual and energy-variance diagnostics, excited-state conditions, the min–max principle, and the generalized Ritz equation $Hc=ESc$.

WKB Quantization uses the classical action. For two simple smooth turning points,

$$\int_{x_1(E_n)}^{x_2(E_n)} p(x;E_n)\,dx = \pi\hbar \left(n+\frac12\right).$$

Its endpoint table distinguishes soft turning points, Dirichlet and Neumann walls, radial Langer corrections, and the Maslov/EBK convention. WKB is an asymptotic estimate, not an upper or lower bound.

The two methods answer different questions. Variational calculations optimize states and give a one-sided ground-energy statement. WKB quantizes classical actions and is usually best for highly excited regular motion.

Scattering observables and constraints

Scattering Cross Section starts from

$$\psi_{\mathbf k}^{(+)}(\mathbf r) \sim e^{i\mathbf k\cdot\mathbf r} + f(\theta,\phi) \frac{e^{ikr}}{r}$$

and gives

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega} = \frac{v_\beta}{v_\alpha} \lvert f_{\beta\alpha}\rvert^2.$$

It distinguishes elastic, reaction, and inclusive total cross sections; partial-wave sums; spin averages; identical-particle event counting; detector acceptance; and long-range caveats.

Born Approximation uses the transform convention

$$\widetilde V(\mathbf q) = \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(\mathbf r)$$

to give

$$f_{\mathrm B}(\mathbf q) = - \frac{\mu}{2\pi\hbar^2} \widetilde V(\mathbf q).$$

The card includes central-potential, Gaussian, Yukawa, scattering-length, and Born phase-shift forms together with validity and perturbative-unitarity tests. The Fourier transform is easy; establishing that repeated scattering is negligible is the substantive part.

Optical Theorem states

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

It identifies $\sigma_{\mathrm{tot}}$ as the sum over all open final channels, derives the elastic and inelastic partial-wave forms, and explains how a real first Born amplitude satisfies unitarity only after the second-order forward imaginary part is included.

Method-selection path

Bound-state energy

1. If an exactly solved Hamiltonian is nearby and the target eigenspace is isolated, start with perturbation theory.
2. If exact degeneracy is present, diagonalize the perturbation in the full degenerate subspace before using nondegenerate formulas.
3. If no controlled small perturbation exists but an admissible state family is available, use the variational bound for the ground state or a min–max/Ritz construction for higher levels.
4. If the classical motion is regular and the action is large compared with $\hbar$, use WKB or EBK with explicitly classified boundaries.
5. Near avoided crossings, threshold states, singular endpoints, or coalescing turning points, use a method adapted to that structure.

Driven transition

1. Compute the finite-time amplitude first.
2. Identify the drive-frequency sign and the energy-conservation condition.
3. Count how many final levels lie inside the width $\hbar/T$.
4. Use the golden rule only when the spectrum is dense on that scale, the matrix element and density vary slowly across the window, and depletion remains small.
5. For strong or coherent few-level driving, use explicit unitary dynamics rather than an irreversible rate.

Scattering problem

1. Fix the asymptotic-state and amplitude convention.
2. Convert amplitudes to flux ratios, retaining outgoing-to-incoming speed factors.
3. If the interaction is weak and short ranged, test the Born approximation rather than assuming it.
4. If the interaction is central, use partial waves to enforce boundary conditions and inspect unitarity channel by channel.
5. Sum elastic and all reaction channels before applying the optical theorem.
6. Treat Coulomb tails, identical particles, spin, and detector acceptance before comparing with observations.

Consistency chains

Several formulas on these cards are designed to check one another.

Perturbative chain

For

$$H(\lambda)=H_0+\lambda V,$$

the coefficients and physical shifts are distinct:

$$E_n(\lambda) = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2E_n^{(2)} + \cdots.$$

A calculation should verify that the dimensionless mixing ratios

$$\left\lvert \frac{ \lambda V_{mn} }{ E_n^{(0)}-E_m^{(0)} } \right\rvert$$

are small outside any subspace treated exactly. A vanishing $E_n^{(1)}$ does not make those ratios small.

Transition-rate chain

Finite-time perturbation theory produces a peak of width

$$\Delta E\sim\frac{\hbar}{T}.$$

The golden-rule delta distribution is justified only after integration against a smooth density of final states. The resulting rate must satisfy a time hierarchy with many final levels resolved statistically but negligible initial-state depletion.

Scattering chain

With a single elastic channel,

$$f(\theta) \longrightarrow \frac{d\sigma}{d\Omega} = \lvert f(\theta)\rvert^2 \longrightarrow \sigma_{\mathrm{el}} = \int d\Omega\, \lvert f\rvert^2.$$

Unitarity then requires

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

When reaction channels open, the angular elastic integral remains $\sigma_{\mathrm{el}}$, while the forward relation returns

$$\sigma_{\mathrm{tot}} = \sigma_{\mathrm{el}} + \sigma_{\mathrm{reac}}.$$

Any mismatch should trigger a normalization, channel-completeness, partial-wave-convergence, or approximation-order audit.

Shared conventions

Symbol	Convention in these cards
$\lambda$	Explicit perturbative bookkeeping parameter unless absorbed into $V$
$\mu$	Reduced mass in a two-body scattering problem
$\mathbf q$	Wave-vector transfer $\mathbf k_f-\mathbf k_i$
$f$	Outgoing spherical-wave coefficient, with dimensions of length
$d\Omega$	Dimensionless solid-angle element $\sin\theta\,d\theta\,d\phi$
$I(E)$	One-way WKB action between endpoints
$J(E)$	Closed-orbit action $\oint p\,dq$
$S_\ell$	Partial-wave scattering element
$a_\ell$	Dimensionless partial amplitude $(S_\ell-1)/(2i)$

Some sources call $\hbar\mathbf q$ the momentum transfer and use $\mathbf q$ for that dimensionful vector. Some include $1/(2\pi)$ in an action variable or distribute $2\pi$ factors through Fourier transforms and $T$ matrices. Translate definitions before comparing results.

Stop signs

Pause before applying a compact formula when any of the following is present:

• an exact or near degeneracy not included in the working subspace;
• a denominator comparable to its coupling matrix element;
• a continuum threshold, shallow bound state, virtual state, or resonance;
• a transition time too short to resolve a continuum or long enough to deplete the source;
• a trial state outside the Hamiltonian or quadratic-form domain;
• a hard, singular, or coalescing WKB endpoint;
• several classically allowed wells connected by tunneling;
• a long-range Coulomb tail or a singular interaction requiring regularization;
• open channels omitted from a cross-section or unitarity sum;
• identical final states counted as distinguishable;
• an approximation being tested against an exact identity at inconsistent perturbative orders.

The symptom-based guide at Common Failure Modes and the evidence hierarchy at Small Parameters and Error Estimates develop these diagnostics.

What each method guarantees

Method	Mathematical status	What it does not guarantee
Rayleigh–Schrödinger perturbation theory	Formal or convergent/asymptotic series under problem-dependent conditions	Accuracy merely because one coefficient is small
Fermi golden rule	Long-time weak-coupling rate after a continuum limit	Exact finite-time dynamics or irreversible decay of a few-level system
Variational principle	Rigorous upper bound for an admissible ground-state trial vector	A lower bound or uniformly accurate observables
Leading WKB	Semiclassical asymptotic quantization	A one-sided spectral bound or universal low-state accuracy
First Born approximation	Leading weak-distortion scattering amplitude	Exact unitarity, resonant behavior, or convergence of the Born series
Optical theorem	Exact unitarity identity in a complete consistently normalized scattering space	Correct dynamics from an amplitude that merely satisfies the identity

Canonical chapters

• Choosing an Approximation Method
• Time-Independent Perturbation Theory
• Time-Dependent Perturbation and Transitions
• Variational and Bound Methods
• WKB and Semiclassical Methods
• Scattering Theory
• Approximation and Semiclassical Methods Formula Sheet

Return to the Formula Compendium for operator, dynamics, canonical-system, spin, density-matrix, and many-body formulas.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, Chs. 16–19.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, Chs. 5–7.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. II, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer, 1999, Ch. 10.