Scattering Amplitude

The scattering amplitude $f(\theta,\phi)$ is the coefficient of the outgoing spherical wave produced by an incoming beam. In three-dimensional elastic potential scattering, a stationary scattering state has the large-distance form

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

The first term is the incident plane wave. The second term is an outgoing spherical wave. The function $f(\theta,\phi)$ tells how much amplitude is scattered into the direction $\hat{\mathbf r}$.

This page is the canonical home for the amplitude convention. Green Function for Scattering derives this coefficient as the far-field transform of the exact interaction source, including normalization and distance conditions. Cross sections are treated in Differential and Total Cross Sections.

Setup

Consider a short-range potential $V(\mathbf r)$ and a particle of mass $m$ with energy

$$E = \frac{\hbar^2k^2}{2m}.$$

Far from the target, where the potential is negligible, the wavefunction solves the free Schrödinger equation. The incoming part is chosen as a plane wave traveling along the incident direction. If the incident beam points along the $z$-axis,

$$e^{i\mathbf k\cdot\mathbf r} = e^{ikz}.$$

The scattered part is outgoing:

$$\frac{e^{ikr}}{r}.$$

The factor $1/r$ is the geometric spreading of a spherical wave in three dimensions.

Dimensions

The wavefunction convention above takes the incoming plane wave to have unit amplitude. Since $e^{ikr}/r$ has dimensions of inverse length, $f$ has dimensions of length:

$$[f]=\text{length}.$$

This is why $|f|^2$ has dimensions of area and can become a differential cross section.

Different normalization conventions can move factors of $2\pi$, $\hbar$, volume, or velocity into related quantities such as the $T$-matrix. The observable cross section is convention-independent when all factors are used consistently.

Relation to Flux

The incident current for a unit-amplitude plane wave is

$$\mathbf j_{\mathrm{inc}} = \frac{\hbar\mathbf k}{m}, \qquad j_{\mathrm{inc}} = \frac{\hbar k}{m}.$$

At large $r$, the radial current carried by the outgoing spherical wave is

$$j_r^{\mathrm{sc}} \approx \frac{\hbar k}{m} \frac{|f(\theta,\phi)|^2}{r^2}.$$

Multiplying by the area element $r^2d\Omega$ gives the scattered flux into solid angle $d\Omega$. Dividing by the incident flux gives

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

This simple relation holds for single-channel elastic scattering with the convention above.

Angular Dependence

For a general target, $f$ can depend on both polar and azimuthal angles:

$$f=f(\theta,\phi).$$

For a central potential $V(r)$, rotational symmetry around the incident beam implies

$$f=f(\theta),$$

with no dependence on $\phi$. In that case, angular momentum methods and Legendre polynomials become natural.

Phase Information

The measured elastic differential cross section determines $|f|^2$, but the amplitude itself is complex:

$$f(\theta,\phi) = |f(\theta,\phi)|e^{i\alpha(\theta,\phi)}.$$

The phase matters for interference, identical-particle scattering, partial-wave unitarity, and the optical theorem. Two amplitudes with the same magnitude can describe different physics when combined with other channels or constraints.

Momentum Transfer

Elastic scattering preserves the magnitude of momentum:

$$|\mathbf k'|=|\mathbf k|=k.$$

The momentum transfer is

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

For elastic scattering,

$$q = 2k\sin\frac{\theta}{2}.$$

Weak-potential scattering amplitudes are often controlled by the Fourier transform of the potential at momentum transfer $\mathbf q$. This is the basis of the Born approximation.

Common Mistakes

• Comparing amplitudes without first checking the normalization convention.
• Treating $f$ as a probability rather than a complex amplitude.
• Forgetting that $|f|^2$ is a differential cross section, not a total cross section.
• Using the three-dimensional spherical-wave form in one-dimensional barrier problems.
• Ignoring phase information when amplitudes interfere.

Exercises

1. Show by dimensional analysis that $|f|^2$ has the dimensions of area.

Solution

In the asymptotic form,

$$\psi\sim e^{ikz}+f\frac{e^{ikr}}{r}.$$

The two terms must have the same dimensions. Since $e^{ikr}$ is dimensionless and $1/r$ has dimensions of inverse length, $f$ must have dimensions of length. Therefore $|f|^2$ has dimensions of length squared, an area.

2. For elastic scattering, derive $q=2k\sin(\theta/2)$.

Solution

Use

$$q^2 = |\mathbf k'-\mathbf k|^2 = k^2+k^2-2k^2\cos\theta = 2k^2(1-\cos\theta).$$

Since $1-\cos\theta=2\sin^2(\theta/2)$,

$$q^2=4k^2\sin^2\frac{\theta}{2},$$

so $q=2k\sin(\theta/2)$.

3. Why does measuring $d\sigma/d\Omega$ not generally determine the full complex amplitude?

Solution

The differential cross section gives $|f(\theta,\phi)|^2$. It loses the phase of $f$. Phase information can reappear through interference, unitarity constraints, spin observables, or comparison among channels, but it is not contained in a single magnitude measurement alone.

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., Springer, 1982.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.