Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

Part VIII, Chapter 3 | Page 3 of 3

Optical Theorem and Advanced Topics

S-matrix unitarity, inelastic scattering, Argand diagrams, Levinson's theorem, and Coulomb scattering

Optical Theorem from Partial Waves

The optical theorem relates the total cross section to the forward scattering amplitude. It can be derived directly from the partial wave expansion. The forward amplitude is:

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

since $P_\ell(1) = 1$ for all $\ell$. Taking the imaginary part:

$$\text{Im}\,f(0) = \sum_{\ell=0}^{\infty}(2\ell+1)\,\frac{\sin^2\delta_\ell}{k}$$

Comparing with the total cross section:

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

This proves the optical theorem:

Optical Theorem
$$\boxed{\sigma_{\text{tot}} = \frac{4\pi}{k}\,\text{Im}\,f(0)}$$

The theorem is a direct consequence of probability conservation: the total flux scattered out of the beam must equal the shadow cast by forward interference between the incident and scattered waves.

The S-Matrix in Partial Waves

The scattering matrix (S-matrix) connects the incoming and outgoing wave amplitudes. For each partial wave, the S-matrix element is:

Partial Wave S-Matrix
$$\boxed{S_\ell = e^{2i\delta_\ell}}$$

The partial wave amplitude is related to the S-matrix by:

$$f_\ell = \frac{S_\ell - 1}{2ik}$$

Unitarity for elastic scattering requires $|S_\ell| = 1$for all $\ell$. This is the mathematical statement that probability is conserved: no flux is created or destroyed, only redirected. In matrix form, unitarity reads $S^\dagger S = \mathbb{1}$.

The unitarity constraint restricts the partial wave amplitude to lie on a circle in the complex plane, known as the unitarity circle:

$$|f_\ell|^2 = \text{Im}\,f_\ell / k \qquad \Leftrightarrow \qquad \left|f_\ell - \frac{i}{2k}\right| = \frac{1}{2k}$$

Inelastic Scattering and Absorption

When inelastic channels are open (particle production, excitation, etc.), flux is lost from the elastic channel. This is described by allowing $|S_\ell| < 1$. We parameterize:

$$S_\ell = \eta_\ell\,e^{2i\delta_\ell}, \qquad 0 \leq \eta_\ell \leq 1$$

where $\eta_\ell$ is the inelasticity parameter($\eta_\ell = 1$ for pure elastic scattering). The elastic, inelastic (reaction), and total cross sections become:

$$\sigma_{\text{el}} = \frac{\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)|1 - \eta_\ell e^{2i\delta_\ell}|^2$$
$$\sigma_{\text{reac}} = \frac{\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)(1 - \eta_\ell^2)$$
$$\sigma_{\text{tot}} = \sigma_{\text{el}} + \sigma_{\text{reac}} = \frac{2\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)(1 - \eta_\ell\cos 2\delta_\ell)$$

Maximum absorption occurs when $\eta_\ell = 0$ (complete absorption of the$\ell$-th partial wave). In this "black disk" limit,$\sigma_{\text{el}} = \sigma_{\text{reac}}$—the elastic and reaction cross sections are equal, a counterintuitive result known as the shadow scatteringphenomenon.

The Argand Diagram

The Argand diagram is a plot of the partial wave amplitude$f_\ell$ in the complex plane (Re $f_\ell$ vs Im $f_\ell$). Decomposing:

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

As the energy varies and $\delta_\ell$ runs from 0 to $\pi$:

  • $f_\ell$ traces a circle of radius $1/(2k)$ centered at $(0, 1/(2k))$
  • The circle starts and ends at the origin
  • At resonance ($\delta_\ell = \pi/2$), $f_\ell$ reaches the top of the circle: $f_\ell = i/k$
  • The speed at which the circle is traced indicates the width of the resonance

With absorption ($\eta_\ell < 1$), the trajectory moves inside the unitarity circle. The Argand diagram provides an intuitive geometric representation of scattering physics and is widely used in particle physics to identify resonances.

Levinson's Theorem

Levinson's theorem establishes a profound connection between scattering data (phase shifts) and bound states. It states:

Levinson's Theorem
$$\boxed{\delta_\ell(0) - \delta_\ell(\infty) = n_b\,\pi}$$

where $n_b$ is the number of bound states with angular momentum $\ell$. The convention is $\delta_\ell(\infty) = 0$ (no phase shift at infinite energy), so $\delta_\ell(0) = n_b\pi$.

This theorem connects the asymptotic behavior of the scattering wavefunction (encoded in phase shifts) with the discrete spectrum (bound states). It follows from the analytic properties of the S-matrix in the complex energy plane: each bound state corresponds to a pole of $S_\ell$ on the positive imaginary $k$-axis, and each pole contributes $\pi$ to the total phase change.

Example: Square Well

A square well of depth $V_0$ and radius $a$ supports$n_b$ s-wave bound states when $(n_b - 1/2)\pi < \sqrt{2mV_0 a^2/\hbar^2} < (n_b + 1/2)\pi$. By Levinson's theorem, $\delta_0(k=0) = n_b\pi$. If a new bound state is about to appear, $\delta_0(0)$ approaches $(n_b+1)\pi$from below, and the scattering length diverges ($|a_s| \to \infty$).

Coulomb Scattering: Logarithmic Phase Shifts

The Coulomb potential presents special difficulties for partial wave analysis because of its infinite range. The asymptotic radial wavefunction acquires a logarithmic phase:

$$u_\ell(r) \xrightarrow{r\to\infty} \sin\!\left(kr - \frac{\ell\pi}{2} - \eta\ln(2kr) + \sigma_\ell\right)$$

where $\eta = Z_1 Z_2 e^2 m/(4\pi\epsilon_0 \hbar^2 k)$ is the Sommerfeld parameter and $\sigma_\ell$ is the Coulomb phase shift:

$$\sigma_\ell = \arg\,\Gamma(\ell + 1 + i\eta)$$

The Coulomb scattering amplitude can be summed exactly to give:

$$f_C(\theta) = -\frac{\eta}{2k\sin^2(\theta/2)}\exp\!\Big[-i\eta\ln\!\big(\sin^2\!\frac{\theta}{2}\big) + 2i\sigma_0\Big]$$

The magnitude gives the Rutherford cross section. When a short-range potential is present in addition to the Coulomb field, one defines nuclear phase shifts$\delta_\ell^{\text{N}}$ relative to the Coulomb baseline, and the scattering amplitude becomes $f = f_C + f_N$.

Connection to Born Approximation at High Energy

At high energies ($ka \gg 1$), many partial waves contribute and the phase shifts become small ($\delta_\ell \ll 1$ for most $\ell$). In this regime, we can approximate $\sin\delta_\ell \approx \delta_\ell$ and$e^{i\delta_\ell} \approx 1$, giving:

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

The Born approximation for the phase shifts can be derived as:

$$\delta_\ell^{(\text{Born})} = -\frac{2mk}{\hbar^2}\int_0^\infty V(r)\,\big[j_\ell(kr)\big]^2\,r^2\,dr$$

Substituting this into the partial wave sum and using the addition theorem for Legendre polynomials recovers the Born scattering amplitude. This demonstrates the consistency of the two approaches and shows that the Born approximation is equivalent to using the leading-order (small) phase shifts.

Key Concepts Summary

  • Partial wave expansion: $f(\theta) = \sum_\ell (2\ell+1)\,f_\ell\,P_\ell(\cos\theta)$ with $f_\ell = e^{i\delta_\ell}\sin\delta_\ell/k$
  • Phase shifts: encode all scattering information; determined by solving the radial equation and matching boundary conditions
  • Low energy: s-wave dominates; scattering length $a = -\lim_{k\to 0}\delta_0/k$ and effective range$r_0$ characterize scattering
  • Breit-Wigner resonance: $\sigma_\ell \propto \Gamma^2/[(E-E_r)^2 + \Gamma^2/4]$ when$\delta_\ell$ passes through $\pi/2$
  • S-matrix: $S_\ell = e^{2i\delta_\ell}$; unitarity requires $|S_\ell| = 1$for elastic scattering
  • Optical theorem: $\sigma_{\text{tot}} = (4\pi/k)\,\text{Im}\,f(0)$ — proved directly from partial wave unitarity
  • Inelastic scattering: $|S_\ell| = \eta_\ell < 1$; reaction cross section$\propto (1-\eta_\ell^2)$
  • Argand diagram: partial wave amplitude traces the unitarity circle; resonances appear as rapid traversal of the circle
  • Levinson's theorem: $\delta_\ell(0) = n_b\pi$ connects phase shifts to bound state count
  • Coulomb scattering: logarithmic phase$\eta\ln(2kr)$ and Coulomb phase shifts $\sigma_\ell = \arg\,\Gamma(\ell+1+i\eta)$
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