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 of approximation symbols, use Common Symbols.

Plane-Wave Normalization

We use time dependence $e^{-iEt/\hbar}$, short-range interactions, and wave-vector states normalized by

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

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 kets obey $|\mathbf p\rangle=\hbar^{-3/2}|\mathbf k=\mathbf p/\hbar\rangle$. This factor ensures $\delta^{(3)}(\mathbf p-\mathbf p\prime)=\hbar^{-3}\delta^{(3)}(\mathbf k-\mathbf k\prime)$.

Scattering Amplitude

The coordinate wavefunction with a unit-amplitude incident wave is a rescaled matrix element:

$$\psi_{\mathbf k}^{(+)}(\mathbf r) \equiv(2\pi)^{3/2}\langle\mathbf r|\psi_{\mathbf k}^{(+)}\rangle.$$

Keeping this factor explicit prevents mixing a delta-normalized ket with a unit incident plane wave. Its large-radius form is

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

For elastic scattering with equal incident and outgoing speeds and distinguishable-particle counting,

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

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. The projectile momentum change is $\hbar\mathbf q$, and the target receives $-\hbar\mathbf q$. Its elastic magnitude is $q=2k\sin(\theta/2)$. 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

Here $m$ is the projectile mass for a fixed target or the reduced mass for relative motion. 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

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

For $H_0=-\hbar^2\nabla^2/(2m)$, the coordinate kernels are

$$\langle\mathbf r|G_0^{(\pm)}|\mathbf r\prime\rangle =-\frac{m}{2\pi\hbar^2}\frac{e^{\pm ik|\mathbf r-\mathbf r\prime|}}{|\mathbf r-\mathbf r\prime|}.$$

The same negative prefactor appears in both kernels. In a spherical-Hankel expansion the prefactors instead differ by complex conjugation: the outgoing $-i$ changes to $+i$ for the incoming kernel. The free Green function derives the source normalization.

S and T Matrices

Define $T(E)=V+VG_0^{(+)}(E)T(E)$. For the wave-vector-normalized kets above,

$$\langle\mathbf k'|S|\mathbf k\rangle =\delta^{(3)}(\mathbf k'-\mathbf k) -2\pi i\,\delta(E_{k'}-E_k) \langle\mathbf k'|T(E_k)|\mathbf k\rangle,$$

and the on-shell amplitude is

$$f(\mathbf k',\mathbf k) =-\frac{4\pi^2m}{\hbar^2} \langle\mathbf k'|T(E_k)|\mathbf k\rangle.$$

The coefficient follows by comparing the outgoing resolvent kernel with the unit-incident wave. It also passes a direct first-order check: $\langle\mathbf k'|V|\mathbf k\rangle=(2\pi)^{-3}\widetilde V(\mathbf q)$ gives precisely the Born formula above. T-Matrix derives these factors and explains on- and off-shell arguments.

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:

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

Relativistic Amplitudes

In this paragraph use $\hbar=c=1$. 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

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

For distinguishable elastic spinless particles with matched external phases, one may define $f=\mathcal M/(8\pi\sqrt{s})$ in this center-of-mass cross-section convention. Spin sums or unequal channel speeds require the complete formula. QFT S-matrix conventions specifies the relativistic states and their assumptions.

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

• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.