← Part VIII/Phase Shifts

4. Phase Shifts

Reading time: ~32 minutes | Pages: 8

Central quantity in scattering theory: encodes all interaction information for each angular momentum.

Physical Meaning

Phase shift $\delta_\ell(k)$: Change in phase of outgoing wave relative to free particle

Attractive potential: $\delta_\ell > 0$ (pulled inward, arrives early)
Repulsive potential: $\delta_\ell < 0$ (pushed outward, arrives late)
No potential: $\delta_\ell = 0$

Complete scattering information contained in $\{\delta_\ell(k)\}$ for all $\ell$

Levinson's Theorem

Relates phase shifts to bound states:

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

where $n_\ell$ is number of bound states with angular momentum $\ell$

Convention: $\delta_\ell(\infty) = 0$, so $\delta_\ell(0) = n_\ell\pi$

Application: Count bound states from low-energy phase shift

Sign of Phase Shift

Attractive potential ($V < 0$):

Wave function pulled inward
Oscillates faster inside potential
$\delta_\ell > 0$
If deep enough: bound states, $\delta_0(0) = n\pi$

Repulsive potential ($V > 0$):

Wave function pushed outward
Oscillates slower inside potential
$\delta_\ell < 0$
No bound states

Energy Dependence

Low energy ($k \to 0$):

S-wave: $\delta_0 \propto k$ (scattering length)
Higher waves: $\delta_\ell \propto k^{2\ell+1}$ (centrifugal suppression)

High energy ($k \to \infty$):

Born approximation valid
$\delta_\ell \to 0$ (short wavelength, minimal distortion)

Resonance:

$\delta_\ell$ passes through $\pi/2$ rapidly
Quasi-bound state (discussed next section)

Example: Hard Sphere

Infinite potential for $r < a$:

$$\delta_\ell = -ka + \arctan\left[\frac{j_\ell(ka)}{n_\ell(ka)}\right]$$

Low-energy limit:

$$\delta_0 \approx -ka, \quad \delta_1 \approx -\frac{(ka)^3}{3}, \quad \delta_2 \approx -\frac{(ka)^5}{15}$$

Higher $\ell$ strongly suppressed

Example: Square Well

Attractive well: $V = -V_0$ for $r < a$

Low-energy s-wave:

$$\tan\delta_0 = \frac{\tan(k'a) - ka}{1 + ka\tan(k'a)}$$

where $k' = \sqrt{k^2 + 2mV_0/\hbar^2}$

Features:

Jumps by $\pi$ each time $k'a = n\pi$ (bound state threshold)
Can have $a_s < 0$ if shallow well

Scattering Length and Effective Range

Low-energy s-wave expansion:

$$k\cot\delta_0 = -\frac{1}{a_s} + \frac{1}{2}r_0 k^2 + \cdots$$

Scattering length $a_s$:

$a_s > 0$: Repulsive scattering (no low-lying bound state)
$a_s < 0$: Attractive with virtual bound state just above threshold
$|a_s| \to \infty$: Bound state exactly at threshold

Effective range $r_0$:

Characteristic size of interaction region
For square well: $r_0 \approx a$

Time Delay

Phase shift related to time delay:

$$\tau_\ell = \frac{2\hbar}{E}\frac{d\delta_\ell}{dE} = \frac{2\hbar}{v}\frac{d\delta_\ell}{dk}$$

Average time particle spends in interaction region

Interpretation:

$\tau_\ell > 0$: Particle delayed (attractive)
$\tau_\ell < 0$: Particle advanced (repulsive)
Resonance: $\tau_\ell$ very large (particle trapped temporarily)

Phase Shift Analysis

Experimental determination:

Measure $d\sigma/d\Omega$ at various angles and energies
Fit to partial wave expansion
Extract phase shifts $\delta_\ell(k)$
Infer potential properties

Challenges:

Multiple $\ell$ contribute at high energy
Uniqueness: different $\delta_\ell$ sets can give same cross section
Polarization measurements help resolve ambiguities

Connection to Bound States

Bound state at energy $E_b < 0$:

Wave function exponentially decaying
Corresponds to pole in S-matrix at imaginary $k$

Analytic continuation:

$$k = i\kappa, \quad E_b = -\frac{\hbar^2\kappa^2}{2m}$$

Phase shift at $E = 0$ remembers bound states below threshold

Applications

Nuclear physics: Nucleon-nucleon scattering reveals nuclear force (strong repulsive core + attractive well)
Atomic collisions: Electron-atom phase shifts determine collision rates
Cold atoms: Scattering length controls interactions in BEC
Particle physics: Meson-nucleon phase shifts reveal resonances
Condensed matter: Impurity scattering, Kondo effect

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