Phase Shifts

A phase shift $\delta_\ell$ measures how a central potential shifts the phase of the $\ell$th radial wave relative to free motion. For short-range elastic scattering,

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

outside the interaction region.

The phase shifts are often the most compact description of central-potential scattering. Once the $\delta_\ell$ are known, the amplitude and cross sections follow.

Phase Relative to Free Motion

The free radial solution behaves asymptotically as

$$u_\ell^{(0)}(r) \sim \sin\left( kr-\frac{\ell\pi}{2} \right).$$

The potential changes this to

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

Thus $\delta_\ell$ is a spatial phase shift relative to the free wave. A packet time delay instead depends on its energy derivative, $\tau_\ell=2\hbar\,d\delta_\ell/dE$, for a sufficiently narrow packet and a fixed free reference. A positive phase shift alone does not guarantee a positive time delay. With this convention, a weak attractive potential tends to give a positive phase shift, while a weak repulsive potential tends to give a negative phase shift.

Relation to the S-Matrix

The operator construction and channel-space meaning of $S$ are developed in S-Matrix. For one-channel elastic scattering,

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

The factor of two appears because the radial scattering matrix compares outgoing and incoming components. Probability conservation implies

$$|S_\ell|=1.$$

If inelastic channels or absorption are present, one often writes

$$S_\ell = \eta_\ell e^{2i\delta_\ell}, \qquad 0\le\eta_\ell\le1.$$

Then $\eta_\ell<1$ measures loss from the elastic channel. Unitarity shows how that loss appears as motion into the partial-wave disk and is balanced by explicit reaction channels.

Relation to the Amplitude

The elastic partial-wave amplitude is

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

Equivalently,

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

The differential cross section is

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

The total elastic cross section is

$$\sigma_{\mathrm{el}} = \frac{4\pi}{k^2} \sum_{\ell=0}^\infty (2\ell+1) \sin^2\delta_\ell.$$

Partial-Wave Cross Sections derives this sum from Legendre orthogonality and extends it to reaction and total channel cross sections. Here the focus is how to interpret and extract phase shifts.

Weak-Potential Estimate

With $m$ the particle mass or reduced mass, a weak central potential gives the first Born estimate

$$\delta_\ell^{\mathrm B} \approx - \frac{2mk}{\hbar^2} \int_0^\infty r^2dr\, V(r) j_\ell^2(kr).$$

This formula is useful as a sign and scale check. It should not be used near resonances, shallow bound states, or other nonperturbative low-energy regimes.

Attractive and Repulsive Potentials

The sign of a phase shift depends on convention, but with the asymptotic convention used here:

• a weak attractive potential $V<0$ tends to produce $\delta_\ell>0$;
• a weak repulsive potential $V>0$ tends to produce $\delta_\ell<0$.

Large phase shifts should be interpreted modulo $\pi$ in observables such as $\sin^2\delta_\ell$. However, the continuous energy dependence of $\delta_\ell(E)$ carries physical information about resonances and bound states.

Levinson Theorem Preview

For short-range potentials under suitable assumptions, Levinson’s theorem relates the zero-energy phase shift to the number of bound states in a partial wave:

$$\delta_\ell(0)-\delta_\ell(\infty) = n_\ell\pi,$$

for regular finite-range potentials with no zero-energy state, taking a phase branch continuous from high energy. A three-dimensional $s$-wave zero-energy resonance contributes an additional $\pi/2$; hard cores and singular potentials require modified statements. The branch choice matters: reducing each phase independently modulo $\pi$ would erase the bound-state count.

Numerical Extraction

To compute $\delta_\ell$ numerically:

1. solve the radial Schrödinger equation outward with the regular origin condition;
2. choose a matching radius outside the range of the potential;
3. match the numerical solution to a combination of free spherical Bessel and Neumann functions;
4. extract the phase from the relative coefficients.

A simpler asymptotic approximation is to match to

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

at large $r$, with $kr$ also large compared with the centrifugal scale. Matching exact free Bessel functions is preferable at modest $kr$; being outside the potential range alone does not justify the sine approximation.

Square-Well Scattering is the exact analytic benchmark for this extraction: matching at one finite radius gives all $\delta_\ell$, exposes the need for phase unwrapping, and produces a near-threshold $p$-wave shape resonance without numerical integration.

Common Mistakes

• Treating $\delta_\ell$ as directly observable without remembering that cross sections use combinations such as $\sin^2\delta_\ell$ and interference sums.
• Forgetting that $S_\ell=e^{2i\delta_\ell}$, not $e^{i\delta_\ell}$.
• Ignoring the energy dependence of phase shifts.
• Using the Born estimate for $\delta_\ell$ at low-energy resonance.
• Extracting phase shifts before the numerical solution has reached the asymptotic region.

Exercises

1. Show that the partial-wave total elastic cross section follows from the orthogonality of Legendre polynomials.

Solution

Use

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

and

$$\int d\Omega\, P_\ell(\cos\theta)P_{\ell'}(\cos\theta) = \frac{4\pi}{2\ell+1} \delta_{\ell\ell'}.$$

Then cross terms vanish and

$$\sigma_{\mathrm{el}} = \int d\Omega\,|f(\theta)|^2 = \frac{4\pi}{k^2} \sum_{\ell=0}^\infty (2\ell+1)\sin^2\delta_\ell.$$

2. If only the $s$-wave contributes, what are $f(\theta)$ and $\sigma_{\mathrm{el}}$?

Solution

For $\ell=0$, $P_0=1$, so

$$f(\theta) = \frac{1}{k} e^{i\delta_0} \sin\delta_0.$$

It is isotropic. The total elastic cross section is

$$\sigma_{\mathrm{el}} = \frac{4\pi}{k^2} \sin^2\delta_0.$$

3. Use the Born phase-shift estimate to determine the sign of $\delta_\ell$ for a weak attractive potential.

Solution

The estimate is

$$\delta_\ell^{\mathrm B} \approx - \frac{2mk}{\hbar^2} \int_0^\infty r^2dr\, V(r) j_\ell^2(kr).$$

For a weak attractive potential, $V(r)<0$ in the interaction region. Since $j_\ell^2(kr)\ge0$, the integral is negative. The overall minus sign makes $\delta_\ell^{\mathrm B}>0$.

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.