Partial-Wave Expansion

Partial-wave expansion solves central-potential scattering by decomposing the wavefunction into angular-momentum channels. For a real, short-range, spin-independent central potential $V(r)$ with one open channel, different $\ell$ values do not mix, and scattering is encoded in one phase shift $\delta_\ell$ per partial wave.

The central result is the elastic scattering amplitude

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

or equivalently

$$f(\theta) = \frac{1}{2ik} \sum_{\ell=0}^\infty (2\ell+1) \left( e^{2i\delta_\ell}-1 \right) P_\ell(\cos\theta).$$

This page explains the expansion. The interpretation and extraction of $\delta_\ell$ are treated in Phase Shifts.

Angular-Momentum Decomposition

For an incident wave along the $z$-axis,

$$e^{ikz} = \sum_{\ell=0}^\infty (2\ell+1)i^\ell j_\ell(kr) P_\ell(\cos\theta),$$

where $j_\ell$ are spherical Bessel functions and $P_\ell$ are Legendre polynomials.

For the ordinary, spherical, and Riccati-Bessel conventions used in radial matching, see Bessel Functions.

For a central potential, the full stationary wavefunction can be written as

$$\psi(\mathbf r) = \sum_{\ell=0}^\infty R_\ell(r) P_\ell(\cos\theta)$$

for an azimuthally symmetric incident beam. More generally, spherical harmonics $Y_\ell^m$ diagonalize angular momentum.

Radial Equation

Use $m$ for the particle mass or reduced mass. Writing

$$R_\ell(r)=\frac{u_\ell(r)}{r},$$

the radial Schrödinger equation is

$$- \frac{\hbar^2}{2m} \frac{d^2u_\ell}{dr^2} + \left[ V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2} \right] u_\ell = Eu_\ell.$$

The term

$$\frac{\hbar^2\ell(\ell+1)}{2mr^2}$$

is the centrifugal barrier. At low energy it suppresses high-$\ell$ penetration into the interaction region.

Asymptotic Radial Form

For a short-range potential, outside the interaction region each radial channel is a free spherical wave with a phase shift:

$$u_\ell(r) \sim A_\ell \sin\left( kr-\frac{\ell\pi}{2}+\delta_\ell \right).$$

The free solution would have $\delta_\ell=0$. The potential changes the phase of the outgoing wave relative to the free radial wave.

Partial-Wave S-Matrix

For elastic scattering in one channel, define

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

Unitarity requires

$$|S_\ell|=1$$

for each partial wave when no absorption or inelastic channel is open. The factor of $2$ appears because $S_\ell$ compares outgoing and incoming radial waves, while $\delta_\ell$ is the standing-wave phase shift.

Unitarity develops the Argand-disk geometry, the $\eta_\ell\lt1$ inelastic case, and elastic and reaction bounds.

Scattering Amplitude

To find the coefficient of each outgoing wave, use $j_\ell(kr)\sim\sin(kr-\ell\pi/2)/(kr)$ in the incident plane-wave expansion. With $x=kr-\ell\pi/2$, the full radial contribution is

$$\frac{(2\ell+1)i^\ell}{2ikr} \left[S_\ell e^{ix}-e^{-ix}\right]P_\ell(\cos\theta).$$

The coefficient of $e^{-ix}$ is fixed by the incident beam. Subtract the free expression, which has $S_\ell=1$: only $(S_\ell-1)e^{ix}$ remains. Since $i^\ell e^{-i\ell\pi/2}=1$, this remainder has precisely the form $f(\theta)e^{ikr}/r$. Thus

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

For elastic scattering with $S_\ell=e^{2i\delta_\ell}$, this becomes

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

This expansion converts a three-dimensional scattering problem into a set of one-dimensional radial problems.

Partial-Wave Cross Sections carries this decomposition through the angular integration and develops elastic, reaction, and total channel sums, threshold scaling, angular-cut interference, and practical truncation tests.

Convergence and Cutoff

For a finite-range potential of range $R$, the largest important angular momenta are roughly

$$\ell_{\max}\sim kR.$$

Semiclassically, $b\simeq(\ell+1/2)/k$ corresponds to an impact parameter. Partial waves with impact parameter much larger than the range of the potential barely interact.

At low energy, the $s$-wave $\ell=0$ often dominates. At higher energy, many partial waves contribute and semiclassical impact-parameter intuition becomes useful. Stationary Phase in Quantum Mechanics shows how saddles of the large-$\ell$ sum recover the deflection function and why coalescing saddles require a rainbow uniform approximation.

Photoelectron Spectroscopy applies partial-wave amplitudes and scattering phases to photoionization thresholds, Cooper minima, and electron angular distributions.

Common Mistakes

• Using partial waves for a noncentral potential without accounting for angular-momentum mixing.
• Forgetting the centrifugal barrier in the radial equation.
• Confusing $\delta_\ell$ with $S_\ell$; they are related by $S_\ell=e^{2i\delta_\ell}$ for elastic scattering.
• Truncating the partial-wave sum without checking convergence.
• Applying short-range formulas directly to unscreened Coulomb scattering.

Exercises

1. Show that if all phase shifts vanish, the partial-wave scattering amplitude is zero.

Solution

Using

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

if $\delta_\ell=0$ for every $\ell$, then $\sin\delta_\ell=0$ for every term. Hence $f(\theta)=0$, as expected for no scattering.

2. Explain why low-energy scattering is often dominated by $\ell=0$.

Solution

The effective radial potential includes the centrifugal barrier

$$\frac{\hbar^2\ell(\ell+1)}{2mr^2}.$$

For $\ell>0$, this barrier suppresses penetration into the short-range interaction region at low energy. The $\ell=0$ channel has no centrifugal barrier, so it usually dominates.

3. If a potential has range $R$, give a semiclassical estimate for the number of important partial waves at wave number $k$.

Solution

The angular momentum is roughly $\ell\sim kb$, where $b$ is the impact parameter. Scattering is important mainly for $b\lesssim R$, so

$$\ell_{\max}\sim kR.$$

Thus the number of contributing partial waves grows with $kR$.

References

• A. Messiah, Quantum Mechanics, Dover, 1999.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.