Quantum Mechanics Course

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Part VIII, Chapter 3 | Page 1 of 3

Partial Wave Expansion

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

Why Partial Waves?

For a spherically symmetric potential $V(r)$, the angular momentum$\ell$ is a good quantum number. Each angular momentum channel scatters independently, and the full scattering problem decomposes into a set of one-dimensional radial equations—one for each value of $\ell$.

Advantages over Born Approximation

  • Exact for spherically symmetric potentials (no perturbative approximation)
  • Works for strong potentials and low energies where Born fails
  • Captures resonances naturally through phase shift behavior
  • Systematic: improve accuracy by including more $\ell$ values
  • At low energy, only a few partial waves contribute ($\ell_{\max}\sim ka$)

Plane Wave Expansion in Partial Waves

The starting point is the expansion of an incident plane wave (propagating along the$z$-axis) in terms of spherical waves. This is the Rayleigh formula:

Rayleigh Plane Wave Expansion
$$\boxed{e^{ikz} = e^{ikr\cos\theta} = \sum_{\ell=0}^{\infty}(2\ell+1)\,i^\ell\,j_\ell(kr)\,P_\ell(\cos\theta)}$$

Here $j_\ell(kr)$ are the spherical Bessel functions of the first kind, and $P_\ell(\cos\theta)$ are the Legendre polynomials. The factor$(2\ell+1)$ is the degeneracy of each angular momentum channel, and$i^\ell$ is a phase factor.

Each term in the sum represents a definite angular momentum $\ell$ with$m = 0$ (azimuthal symmetry about the beam axis). The spherical Bessel functions have the asymptotic form:

$$j_\ell(kr) \xrightarrow{r\to\infty} \frac{\sin(kr - \ell\pi/2)}{kr} = \frac{e^{i(kr-\ell\pi/2)} - e^{-i(kr-\ell\pi/2)}}{2ikr}$$

This represents equal incoming and outgoing spherical waves—no net flux in any partial wave, as expected for a free particle.

Effect of the Potential: Phase Shifts

When a spherically symmetric potential is present, the radial wavefunction for each$\ell$ is modified. Outside the range of the potential ($r > a$), the radial equation reduces to the free equation, and the general solution is:

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

The only effect of the potential on the asymptotic wavefunction is to introduce a phase shift $\delta_\ell$ for each partial wave. This is a remarkable simplification: all the information about scattering from a spherical potential is encoded in the set of numbers $\{\delta_0, \delta_1, \delta_2, \ldots\}$.

Without the potential, $\delta_\ell = 0$ for all $\ell$. An attractive potential ($V < 0$) "pulls in" the wavefunction, producing positive phase shifts. A repulsive potential ($V > 0$) "pushes out" the wavefunction, producing negative phase shifts.

Scattering Amplitude from Phase Shifts

Comparing the asymptotic form of the full wavefunction with the standard scattering form $\psi \sim e^{ikz} + f(\theta)\,e^{ikr}/r$, and using the partial wave expansion, one obtains:

Partial Wave Expansion of Scattering Amplitude
$$\boxed{f(\theta) = \sum_{\ell=0}^{\infty}(2\ell+1)\,f_\ell\,P_\ell(\cos\theta)}$$

where the partial wave amplitude $f_\ell$ is:

Partial Wave Amplitude
$$\boxed{f_\ell = \frac{e^{2i\delta_\ell} - 1}{2ik} = \frac{1}{k}\,e^{i\delta_\ell}\sin\delta_\ell}$$

The two forms are equivalent. The first makes the connection to the $S$-matrix transparent ($S_\ell = e^{2i\delta_\ell}$), while the second is useful for direct calculation.

Total Cross Section

The differential cross section is $d\sigma/d\Omega = |f(\theta)|^2$. Integrating over solid angles and using the orthogonality of Legendre polynomials:

$$\int_0^\pi P_\ell(\cos\theta)\,P_{\ell'}(\cos\theta)\,\sin\theta\,d\theta = \frac{2}{2\ell+1}\,\delta_{\ell\ell'}$$

the cross terms vanish and we obtain:

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

Each partial wave contributes independently to the total cross section. The maximum contribution from a single partial wave occurs when $\delta_\ell = \pi/2 + n\pi$(a resonance), giving the unitarity limit:

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

No partial wave can scatter more than this amount—a direct consequence of probability conservation (unitarity).

The Radial Schrodinger Equation

The radial equation that determines the phase shifts is (with $u_\ell(r) = r\,R_\ell(r)$):

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

The effective potential includes the centrifugal barrier$\hbar^2\ell(\ell+1)/(2mr^2)$, which grows with $\ell$ and prevents low-energy particles from reaching the interaction region. This is why only low$\ell$ partial waves contribute at low energy.

The boundary conditions are:

  • $u_\ell(0) = 0$ (regularity at the origin)
  • $u_\ell(r) \to \sin(kr - \ell\pi/2 + \delta_\ell)$ as $r\to\infty$

The procedure is: integrate the radial equation outward from $r = 0$, then match the solution to the asymptotic form at large $r$ to extract$\delta_\ell$. In practice, only $\ell_{\max} \sim ka$ partial waves need to be computed, since higher partial waves have $\delta_\ell \approx 0$.

Semiclassical Picture of Partial Waves

A semiclassical argument reveals why only $\ell \lesssim ka$ partial waves contribute. Classically, a particle with angular momentum $L = \ell\hbar$passes the scattering center at an impact parameter:

$$b = \frac{L}{p} = \frac{\ell\hbar}{\hbar k} = \frac{\ell}{k}$$

If the potential has range $a$, only particles with $b \lesssim a$interact with it. Therefore only $\ell \lesssim ka$ partial waves are appreciably scattered. Higher partial waves "miss" the potential entirely, and their phase shifts are exponentially small.

This insight also explains the centrifugal barrier suppression quantitatively: the height of the barrier for partial wave $\ell$ is approximately$E_{\text{barrier}} \sim \hbar^2\ell^2/(2ma^2)$. A particle with energy$E = \hbar^2 k^2/(2m)$ can classically overcome this barrier only if$\ell \lesssim ka$.

Truncation of the Partial Wave Sum

Practical Guidelines

  • Low energy ($ka \ll 1$): Only s-wave ($\ell = 0$) matters. Scattering is isotropic.
  • Moderate energy ($ka \sim 1$): A few partial waves suffice ($\ell = 0, 1, 2$). Angular distributions show structure.
  • High energy ($ka \gg 1$): Many partial waves needed; sum can be replaced by an integral (semiclassical limit). Born approximation is usually simpler and equivalent.
  • Rule of thumb: include up to $\ell_{\max} = ka + 4\sqrt{ka} + 4$for convergence to within 1%.

Interference Between Partial Waves

Although each partial wave contributes independently to the total cross section (due to Legendre polynomial orthogonality), the partial waves interfere in the differential cross section:

$$\frac{d\sigma}{d\Omega} = |f(\theta)|^2 = \left|\sum_\ell (2\ell+1) f_\ell P_\ell(\cos\theta)\right|^2$$

This interference produces the rich angular structure observed in scattering experiments: forward peaks, backward peaks, and diffraction minima. The positions of these features encode information about the phase shifts and hence the potential.

For example, s-wave and p-wave interference ($\ell = 0$ and $\ell = 1$) produces a forward-backward asymmetry proportional to $\cos\theta$, since$P_0(\cos\theta) = 1$ and $P_1(\cos\theta) = \cos\theta$.

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