← Part VIII/Partial Wave Analysis

3. Partial Wave Analysis

Reading time: ~40 minutes | Pages: 10

Exact method for spherical potentials: decompose scattering into angular momentum channels.

Why Partial Waves?

Advantages over Born approximation:

Exact for spherically symmetric potentials
Works for strong potentials and low energies
Captures resonances naturally
Systematic: improve by including more $\ell$ values

Key idea: Each angular momentum $\ell$ scatters independently

Expansion in Spherical Waves

For spherical potential $V(r)$, expand wave function:

$$\psi(\vec{r}) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell R_\ell(r)Y_\ell^m(\theta,\phi)$$

For scattering along z-axis, only $m = 0$ contributes:

$$\psi(r,\theta) = \sum_{\ell=0}^\infty \frac{u_\ell(r)}{r}P_\ell(\cos\theta)$$

Radial equation for each $\ell$ decouples

Radial Schrödinger Equation

For each partial wave $\ell$:

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

Effective potential includes centrifugal barrier

Boundary conditions:

$u_\ell(0) = 0$ (regularity at origin)
Asymptotic form matches incoming + outgoing waves

Asymptotic Behavior

For $r \to \infty$ (outside potential range):

$$u_\ell(r) \xrightarrow{r\to\infty} \sin(kr - \ell\pi/2 + \delta_\ell)$$

where $\delta_\ell$ is the phase shift

Without potential ($V = 0$):

$$u_\ell^{(0)}(r) = \sin(kr - \ell\pi/2)$$

Phase shift $\delta_\ell$: Change in phase due to scattering

Scattering Amplitude

Partial wave expansion:

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

Alternative form:

$$f(\theta) = \sum_{\ell=0}^\infty (2\ell+1)f_\ell P_\ell(\cos\theta)$$

where $f_\ell = \frac{e^{i\delta_\ell}\sin\delta_\ell}{k} = \frac{e^{2i\delta_\ell}-1}{2ik}$ is partial wave amplitude

Partial Wave Cross Sections

Cross section for $\ell$-th partial wave:

$$\sigma_\ell = \frac{4\pi}{k^2}(2\ell+1)\sin^2\delta_\ell$$

Total cross section:

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

Unitarity limit:

$$\sigma_\ell \leq \frac{4\pi}{k^2}(2\ell+1)$$

Maximum when $\delta_\ell = \pi/2$ (resonance)

Low Energy (s-wave) Scattering

For $ka \ll 1$ (wavelength $\gg$ range):

Centrifugal barrier suppresses $\ell > 0$
Only $\ell = 0$ (s-wave) contributes

S-wave scattering length:

$$a_s = -\lim_{k\to 0}\frac{\tan\delta_0}{k} = -\lim_{k\to 0}\frac{\delta_0}{k}$$

Low-energy cross section:

$$\sigma = 4\pi a_s^2$$

Independent of energy, isotropic

Hard Sphere Scattering

Infinite potential for $r < a$, zero for $r > a$:

Boundary condition: $u_\ell(a) = 0$

Phase shifts:

$$\tan\delta_\ell = \frac{j_\ell(ka)}{n_\ell(ka)}$$

where $j_\ell, n_\ell$ are spherical Bessel and Neumann functions

Low-energy limit:

$$\delta_0 \approx -ka, \quad a_s = a, \quad \sigma = 4\pi a^2$$

Square Well Scattering

Potential: $V = -V_0$ for $r < a$, zero otherwise

Inside well: wave number $k' = \sqrt{k^2 + 2mV_0/\hbar^2}$

Matching at $r = a$ gives:

$$\tan\delta_\ell = \frac{kj_\ell'(ka)j_\ell(k'a) - k'j_\ell(ka)j_\ell'(k'a)}{kn_\ell'(ka)j_\ell(k'a) - k'n_\ell(ka)j_\ell'(k'a)}$$

Features:

Resonances when $\delta_\ell \approx \pi/2$
Can have negative scattering length (virtual bound state)

Ramsauer-Townsend Effect

Minimum in scattering cross section at specific energy

Condition: $\delta_0 = n\pi$ for integer $n$

Partial transparency: incident wave passes through without scattering
Observed in electron scattering from noble gases
Purely quantum effect (no classical analog)

Effective Range Theory

Low-energy expansion of $\delta_0$:

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

where $r_0$ is effective range

Parameters encode potential:

$a_s$: scattering length
$r_0$: characteristic range of potential

Application: Nucleon-nucleon scattering

Argand Diagram

Plot partial wave amplitude in complex plane:

$$f_\ell = \frac{e^{i\delta_\ell}\sin\delta_\ell}{k} = \frac{\cos\delta_\ell \sin\delta_\ell + i\sin^2\delta_\ell}{k}$$

As energy varies:

Traces circle of radius 1/(2k)
Centered at (0, 1/(2k))
Resonance: rapid change in $\delta_\ell$, circle traced quickly

Coulomb + Short-Range Potential

For $V = V_C(r) + V_{SR}(r)$:

Asymptotic form includes Coulomb phase:

$$u_\ell(r) \xrightarrow{r\to\infty} \sin(kr - \ell\pi/2 - \eta\ln(2kr) + \sigma_\ell + \delta_\ell^{SR})$$

where $\eta$ is Sommerfeld parameter, $\sigma_\ell$ is Coulomb phase

$\delta_\ell^{SR}$ isolates effect of short-range part

Computational Approach

Procedure:

Numerically integrate radial equation from $r = 0$
Match to asymptotic form at large $r$
Extract phase shift $\delta_\ell$
Calculate cross section from phase shifts
Repeat for all needed $\ell$ values

Truncation: Sum converges, typically need $\ell_{max} \sim ka + \text{few}$

Higher $\ell$: centrifugal barrier prevents interaction, $\delta_\ell \approx 0$

← Born Approximation Phase Shifts →

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