Phase Shifts and Resonances

Calculating Phase Shifts: Boundary Matching

The phase shifts are determined by solving the radial Schrodinger equation inside the potential region and matching to the asymptotic free-particle solution at the boundary. For a potential that vanishes for $r > a$, the solution in the exterior region is a linear combination of spherical Bessel and Neumann functions:

The matching conditions at $r = a$ require continuity of $u_\ell$and $u_\ell'$. Defining the logarithmic derivative:

the phase shift is determined by matching logarithmic derivatives across the boundary:

Here primes denote derivatives with respect to the argument. The interior logarithmic derivative $\gamma_\ell$ is obtained by integrating the radial equation from $r = 0$ to $r = a$.

Hard Sphere Scattering

The simplest example is the hard sphere: $V = \infty$ for $r < a$and $V = 0$ for $r > a$. The boundary condition is$u_\ell(a) = 0$, which gives:

For low energies ($ka \ll 1$), using the small-argument expansions of the spherical Bessel functions:

we find that the phase shifts vanish rapidly with increasing $\ell$:

This confirms the dominance of s-wave ($\ell = 0$) scattering at low energy. For the hard sphere specifically: $\delta_0 \approx -ka$, so the scattering length is $a_s = a$ and $\sigma = 4\pi a^2$.

Low-Energy Limit: s-Wave Dominance

When the de Broglie wavelength is much larger than the range of the potential ($ka \ll 1$), the centrifugal barrier $\sim \ell(\ell+1)/r^2$prevents higher partial waves from reaching the scattering center. Only the s-wave ($\ell = 0$) contributes significantly:

The scattering is isotropic (independent of angle) since $P_0(\cos\theta) = 1$.

Scattering Length

The scattering length $a$ is defined as the low-energy limit of the phase shift:

The low-energy cross section is:

The sign of the scattering length carries physical meaning:

• $a > 0$: effectively repulsive interaction (e.g., hard sphere gives $a = R$)
• $a < 0$: effectively attractive interaction (e.g., neutron-proton singlet state)
• $|a| \to \infty$: near a zero-energy bound state or virtual state (unitarity limit)

Effective Range Expansion

The low-energy behavior of the s-wave phase shift is captured systematically by the effective range expansion:

Here $a$ is the scattering length and $r_0$ is the effective range, which characterizes the spatial extent of the potential. This expansion is remarkably universal—it depends on only two parameters regardless of the detailed shape of $V(r)$.

For the hard sphere of radius $R$: $a = R$ and $r_0 = 2R/3$. For nucleon-nucleon scattering, measured values are $a \approx -23.7$ fm (singlet) and $a \approx 5.4$ fm (triplet), with $r_0 \approx 2.7$ fm for both.

Breit-Wigner Resonances

A scattering resonance occurs when a phase shift passes rapidly through$\pi/2$ (or $\pi/2 + n\pi$) as the energy increases. Near a resonance at energy $E_r$, the phase shift has the characteristic form:

where $\Gamma$ is the resonance width and$\delta_\ell^{(\text{bg})}$ is a slowly varying background phase. The partial wave cross section near a resonance takes the Breit-Wigner form:

This is a Lorentzian peak centered at $E = E_r$ with full width at half maximum$\Gamma$. At the peak ($E = E_r$), the cross section reaches the unitarity limit $4\pi(2\ell+1)/k^2$.

The Ramsauer-Townsend Effect

The Ramsauer-Townsend effect is a striking quantum phenomenon in which the scattering cross section drops to nearly zero at a specific energy. It occurs when the s-wave phase shift passes through $\delta_0 = n\pi$ (integer $n$), making $\sin^2\delta_0 = 0$.

If higher partial waves are negligible (low energy), the total cross section vanishes:

The incoming wave passes through the potential region with its phase shifted by exactly a multiple of $\pi$, emerging as if no scattering occurred. This is partial transparency—a purely quantum effect with no classical analog.

Square Well Phase Shifts

For the finite square well $V = -V_0$ for $r < a$, zero otherwise, the interior wave number is $k' = \sqrt{k^2 + 2mV_0/\hbar^2}$. Matching the interior and exterior solutions at $r = a$ gives the phase shifts:

For the s-wave ($\ell = 0$), this simplifies to:

Key features:

• Resonances appear when $\delta_\ell$ passes through $\pi/2$, corresponding to quasi-bound states in the well
• Negative scattering length indicates a virtual bound state (pole of the S-matrix on the negative imaginary $k$-axis)
• Bound states exist when $\delta_\ell(k=0) = n\pi$ with $n \geq 1$ (Levinson's theorem)

Shape Resonances vs. Feshbach Resonances

There are two distinct types of resonances in quantum scattering:

The Breit-Wigner formula applies to both types near an isolated resonance. The width$\Gamma$ has a characteristic energy dependence:

Higher partial wave resonances are inherently narrower at low energy because the centrifugal barrier suppresses tunneling.