Phase Shift Extraction Notebook

This notebook extracts elastic partial-wave phase shifts by solving the radial Schrodinger equation and matching the numerical solution to known free asymptotic functions. It turns the formal definition of $\delta_\ell$ into a reproducible scattering calculation.

Physical Problem

Use a short-range central potential, for example

$$V(r)=V_0e^{-r^2/a^2}.$$

For reduced mass $\mu$ and energy

$$E=\frac{\hbar^2k^2}{2\mu},$$

the reduced radial wavefunction obeys

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

The regular origin behavior is

$$u_\ell(r)\propto r^{\ell+1}$$

as $r\to0$.

Dimensionless Form

Set the range $a$ as the unit of length:

$$\rho=\frac{r}{a}.$$

Use the dimensionless energy and strength

$$\epsilon=k^2a^2, \qquad \nu=\frac{2\mu V_0a^2}{\hbar^2}.$$

Then the radial equation becomes

$$- \frac{d^2u_\ell}{d\rho^2} + \left[ \nu e^{-\rho^2} + \frac{\ell(\ell+1)}{\rho^2} \right] u_\ell = \epsilon u_\ell.$$

This form makes parameter sweeps independent of units.

Numerical Setup

For each $\ell$ and $k$:

1. choose a small starting radius $\rho_{\min}$;
2. initialize the regular solution with $u_\ell\propto\rho^{\ell+1}$;
3. integrate outward to a matching radius $\rho_m$ where $V(\rho_m)$ is negligible;
4. compute the logarithmic derivative

$$L = \frac{u_\ell'(r_m)}{u_\ell(r_m)};$$

5. match to free spherical solutions to extract $\delta_\ell$.

Numerov integration is a common choice for this second-order equation. A high-order adaptive ODE solver can also be used if the normalization and phase are handled consistently.

Matching Formula

Outside the potential range, write the reduced radial solution as

$$u_\ell(r) = C \left[ \hat j_\ell(kr)\cos\delta_\ell - \hat n_\ell(kr)\sin\delta_\ell \right],$$

where

$$\hat j_\ell(z)=zj_\ell(z), \qquad \hat n_\ell(z)=zn_\ell(z)$$

are Riccati-Bessel functions.

The conventions for $j_\ell$, $n_\ell$, and their Riccati forms are summarized in Bessel Functions.

Let

$$z=kr_m, \qquad J=\hat j_\ell(z), \qquad N=\hat n_\ell(z),$$

and let $J'$ and $N'$ denote derivatives with respect to $z$. The logarithmic derivative matching gives

$$\tan\delta_\ell = \frac{kJ'-LJ}{kN'-LN}.$$

This formula is insensitive to the arbitrary normalization of the numerical solution.

Parameter Sweep

A useful first sweep is:

$$\ell=0,1,2, \qquad ka=0.1,0.3,1,3, \qquad \nu=-0.5,-2,-5.$$

At weak coupling, compare the numerical phase shifts with the Born-estimate sign and scale. At low $k$, verify $s$-wave dominance unless the potential is tuned near a threshold state.

Expected Figures

The notebook should generate:

• $\delta_\ell(k)$ versus $k$ for several $\ell$;
• $k\cot\delta_0(k)$ versus $k^2$ for extracting the scattering length and effective range;
• convergence of $\delta_\ell$ with matching radius;
• convergence with radial step size;
• optionally, the numerical radial wavefunction and its matched asymptotic form.

Phase shifts should be unwrapped or plotted with branch choices stated clearly.

Validation Checks

For $V=0$, the extracted phase shifts must be

$$\delta_\ell=0$$

up to numerical error.

For sufficiently low energy and short-range potential, check the threshold behavior

$$\delta_\ell(k)\propto k^{2\ell+1}.$$

For $\ell=0$, extract the scattering length from

$$a = - \lim_{k\to0} \frac{\tan\delta_0(k)}{k}.$$

The total elastic cross section computed from phase shifts,

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

should stabilize as $\ell_{\max}$ increases.

Before using a smooth potential, validate the implementation against Square-Well Scattering and Hard-Sphere Scattering. The first checks interface matching, scattering-length extrapolation, and resolution of a narrow $p$-wave feature. The second isolates the exterior boundary condition and tests partial-wave cutoff convergence from threshold to $kR\gg1$. Low-Energy S-Wave Scattering then tests extraction of $k\cot\delta_0$, the effective range, the unitarity window, and a shallow pole. Resonance from a Square Well tests phase unwrapping, background-aware fitting, and continuation to an outgoing $p$-wave pole.

Born Approximation Numerical Test specializes the same matching idea to a repulsive Gaussian and supplies a complete convergence ledger before using the phases to judge first Born accuracy.

Known Failure Modes

• Matching before the potential is negligible.
• Taking $u_\ell'/u_\ell$ near a node of the numerical solution.
• Using ordinary Bessel functions where Riccati-Bessel functions are required.
• Losing phase-shift branch continuity as $k$ varies.
• Applying short-range matching formulas to a Coulomb tail.
• Under-resolving oscillatory wavefunctions at large $k$.

Reproducibility Metadata

Record:

• potential form and dimensionless strength;
• mass and unit conventions;
• grid spacing and radial domain;
• ODE solver and tolerances;
• matching radius;
• Bessel-function library;
• phase branch convention;
• convergence criteria for $\ell_{\max}$ and radial step size.

Cross-Links

• Computational Notebooks
• Phase Shifts
• Partial-Wave Expansion
• Bessel Functions
• Low-Energy Scattering
• Scattering Length
• Square-Well Scattering
• Hard-Sphere Scattering
• Gaussian Potential in the Born Approximation
• Yukawa Potential in the Born Approximation
• Low-Energy S-Wave Scattering
• Resonance from a Square Well

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.
• B. R. Johnson, “The multichannel log-derivative method for scattering calculations,” Journal of Computational Physics 13, 445-449, 1973.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.

Exercises

1. Why does logarithmic-derivative matching avoid dependence on the arbitrary normalization of $u_\ell$?

Solution

If $u_\ell$ is multiplied by a constant $C$, then both $u_\ell'$ and $u_\ell$ are multiplied by $C$. Their ratio

$$L=\frac{u_\ell'}{u_\ell}$$

is unchanged. Since the matching formula uses $L$, the extracted phase shift is independent of the normalization chosen during integration.

2. Why should the free-potential test give $\delta_\ell=0$?

Solution

When $V=0$, the regular radial solution is proportional to the free Riccati-Bessel function $\hat j_\ell(kr)$. In the matching form, this corresponds to $\cos\delta_\ell=1$ and $\sin\delta_\ell=0$, hence $\delta_\ell=0$ modulo $\pi$. The branch connected continuously to zero potential is $\delta_\ell=0$.