Scattering Convention Dictionary

Scattering formulas are unusually sensitive to convention. Factors of $2\pi$, $\hbar$, energy delta functions, state normalization, and the sign of the outgoing Green function can move between definitions while leaving physical cross sections unchanged.

This dictionary fixes the conventions used in this volume and flags common alternatives. When comparing two sources, compare complete observables, not isolated symbols.

For perturbative gaps, state corrections, variational notation, semiclassical actions, and symbol-collision rules, see Notation and Conventions. For a compact lookup spanning the whole volume, use Common Symbols.

Plane-Wave Normalization

This volume often uses wave-number states with

$$\langle \mathbf r|\mathbf k\rangle = \frac{1}{(2\pi)^{3/2}} e^{i\mathbf k\cdot\mathbf r},$$

so that

$$\langle \mathbf k|\mathbf k'\rangle = \delta^{(3)}(\mathbf k-\mathbf k').$$

Momentum states may instead be normalized with

$$\langle \mathbf r|\mathbf p\rangle = \frac{1}{(2\pi\hbar)^{3/2}} e^{i\mathbf p\cdot\mathbf r/\hbar},$$

and

$$\langle \mathbf p|\mathbf p'\rangle = \delta^{(3)}(\mathbf p-\mathbf p').$$

Since $\mathbf p=\hbar\mathbf k$, the two conventions shift powers of $\hbar$ between states, delta functions, and amplitudes.

Scattering Amplitude

The central potential-scattering convention used here is

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

With this convention,

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

The amplitude $f$ has dimensions of length. It is not the same object as the relativistic invariant amplitude $\mathcal M$ used in QFT.

Momentum Transfer

For elastic scattering with incoming wave vector $\mathbf k$ and outgoing wave vector $\mathbf k'$,

$$\mathbf q = \mathbf k'-\mathbf k$$

is the wave-vector transfer. Some sources define the opposite sign. For central potentials the Born amplitude depends on $|\mathbf q|$, so the sign is often invisible. For phases, noncentral potentials, or external-field conventions, the sign must be checked.

Born Amplitude

With the asymptotic convention above, the first Born amplitude is

$$f_{\mathrm B}(\mathbf q) = - \frac{m}{2\pi\hbar^2} \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(\mathbf r).$$

If another source uses a different plane-wave normalization or defines $\mathbf q=\mathbf k-\mathbf k'$, the sign in the Fourier phase and the prefactor may look different.

Green Function Prescription

The outgoing and incoming free resolvents are

$$G_0^{(+)}(E) = \frac{1}{E-H_0+i0}, \qquad G_0^{(-)}(E) = \frac{1}{E-H_0-i0}.$$

The $+i0$ prescription gives outgoing spherical waves in the asymptotic state

$$|\psi^{(+)}\rangle = |\phi\rangle + G_0^{(+)}(E)V|\psi^{(+)}\rangle.$$

Some texts use advanced or retarded terminology in ways that depend on Fourier-transform sign conventions. Check the resulting boundary condition, not only the symbol.

S and T Matrices

S-Matrix fixes the asymptotic operator-level meaning. T-Matrix develops the transition operator, exact amplitude factors, and shell structure. A common energy-normalized convention is

$$S_{fi} = \delta_{fi} - 2\pi i\, \delta(E_f-E_i) T_{fi}.$$

Another common schematic convention is

$$S=I+i\mathcal T.$$

Both are useful, but the object called $T$ or $\mathcal T$ differs by delta functions, signs, and normalization factors. When deriving the optical theorem, use one convention consistently from the start.

Partial Waves

For one-channel elastic central scattering,

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

The scattering amplitude is

$$f(\theta) = \frac{1}{2ik} \sum_{\ell=0}^\infty (2\ell+1) (S_\ell-1) P_\ell(\cos\theta).$$

Equivalently,

$$f(\theta) = \frac1k \sum_{\ell=0}^\infty (2\ell+1) e^{i\delta_\ell} \sin\delta_\ell P_\ell(\cos\theta).$$

The factor of two in $S_\ell=e^{2i\delta_\ell}$ is a frequent source of errors.

Optical Theorem

With the amplitude convention above,

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

If the forward amplitude is defined with different factors, the optical theorem changes accordingly. The invariant content is unitarity:

$$S^\dagger S=I.$$

QFT Normalization Warning

Relativistic QFT usually defines an invariant amplitude $\mathcal M$ through an $S$-matrix element with four-momentum delta functions and relativistically normalized states. A common two-to-two formula is

$$\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2s} \frac{|\mathbf p_f|}{|\mathbf p_i|} \overline{|\mathcal M|^2}.$$

Do not identify $f$ and $\mathcal M$ directly. They live in different normalization systems.

Comparison Checklist

When comparing scattering formulas, check:

• whether states are normalized in $\mathbf k$, $\mathbf p$, energy, or relativistic phase space;
• whether $S=I+iT$ or $S=I-2\pi i\delta(E)T$ is used;
• whether $\mathbf q$ is incoming minus outgoing or outgoing minus incoming;
• whether $G_0^{(+)}$ means outgoing waves with the Fourier convention used;
• whether phase shifts enter as $S_\ell=e^{2i\delta_\ell}$;
• whether the amplitude is $f$, $T$, $\mathcal T$, or $\mathcal M$;
• whether identical-particle, spin, or channel factors are included.

Exercises

1. Why can two Born-amplitude formulas differ by a sign in the Fourier phase and still agree physically for a central real potential?

Solution

For a central real potential, the Fourier transform depends only on $q=|\mathbf q|$ and is even under $\mathbf q\to-\mathbf q$. Therefore conventions using $e^{-i\mathbf q\cdot\mathbf r}$ and $e^{+i\mathbf q\cdot\mathbf r}$ give the same function of $q$ in that special case. For noncentral or complex interactions, the sign convention must be tracked.

2. In one-channel elastic scattering, why is $S_\ell=e^{2i\delta_\ell}$ rather than $e^{i\delta_\ell}$?

Solution

The phase shift $\delta_\ell$ is the shift of a standing radial wave relative to the free sine wave. The $S$-matrix compares outgoing and incoming radial wave components. The outgoing component gains the opposite relative phase compared with the incoming component, so the ratio carries twice the standing-wave phase shift:

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

References

• 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.
• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.