Partial-wave analysis

Partial wave analysis, in the context of quantum mechanics, refers to a technique for solving scattering problems by decomposing each wave into its constituent angular momentum components and solving using boundary conditions.

Partial wave expansion

In the standard scattering problem, the incoming beam is assumed to take the form of a plane wave of wave number k, which can be decomposed into partial waves using the plane wave expansion in terms spherical Bessel functions and Legendre polynomials:

$\psi _{\text{in}}(\mathbf {r} )=e^{ikz}=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta )$

Here, we have assumed a spherical coordinate system in which the z-axis is aligned with the beam direction. The radial part of the this wave function consists solely of the spherical Bessel function, which can be rewritten as a sum of two spherical Hankel functions:

$j_{\ell }(kr)={\frac {1}{2}}\left(h_{\ell }^{(1)}(kr)+h_{\ell }^{(2)}(kr)\right)$

This has physical significance: h_ℓ⁽¹⁾ asymptotically (i.e. for large r) behaves as i^−(ℓ+1)e^ikr/(kr) and is thus an outgoing wave, whereas h_ℓ⁽²⁾ asymptotically behaves as i^ℓ+1e^−ikr/(kr) and is thus an incoming wave. The incoming wave is unaffected by the scattering, while the outgoing wave is modified by a factor known as the partial wave S-matrix element S_ℓ:

${\frac {u_{\ell }(r)}{r}}{\stackrel {r\to \infty }{\longrightarrow }}{\frac {i^{\ell }k}{\sqrt {2\pi }}}\left(h_{\ell }^{(1)}(kr)+S_{\ell }h_{\ell }^{(2)}(kr)\right)$

where u_ℓ(r)/r is the radial component of the actual wave function. The scattering phase shift δ_ℓ is defined as half of the phase of S_ℓ:

$S_{\ell }=e^{2i\delta _{\ell }}$

If flux is not lost, then |S_ℓ| = 1 and thus the phase shift is real. This is typically the case unless the potential has an imaginary absorptive component, which is often used in phenomenological models to simulate loss due to other reaction channels.

Therefore, the full wave function is, asymptotically,

$\psi (\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }{\frac {h_{\ell }^{(1)}(kr)+S_{\ell }h_{\ell }^{(2)}(kr)}{2}}P_{\ell }(\cos \theta )$

Subtracting ψ_in yields the asymptotic outgoing wave function:

$\psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }{\frac {S_{\ell }-1}{2}}h_{\ell }^{(2)}(kr)P_{\ell }(\cos \theta )$

Making use of the asymptotic behavior of the spherical Hankel functions, one obtains:

$\psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}{\frac {e^{ikr}}{r}}\sum _{\ell =0}^{\infty }(2\ell +1){\frac {S_{\ell }-1}{2ik}}P_{\ell }(\cos \theta )$

Since the scattering amplitude f(θ, φ) is defined via:

$\psi _{\text{out}}(\mathbf {r} ){\stackrel {r\to \infty }{\longrightarrow }}{\frac {e^{ikr}}{r}}f(\theta ,\phi )$

It follows that

$f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }(2\ell +1){\frac {S_{\ell }-1}{2ik}}P_{\ell }(\cos \theta )=\sum _{\ell =0}^{\infty }(2\ell +1){\frac {e^{i\delta }\sin \delta }{k}}P_{\ell }(\cos \theta )$

and thus the differential cross section is given by

${\frac {d\sigma }{d\Omega }}=|f(\theta ,\varphi )|^{2}={\frac {1}{k^{2}}}\left|\sum _{\ell =0}^{\infty }(2\ell +1)e^{i\delta _{\ell }}\sin \delta _{\ell }P_{\ell }(\cos \theta )\right|^{2}$

This works for any short-ranged interaction. For long-ranged interactions (such as the Coulomb interaction), the summation over ℓ may not converge. The general approach for such problems to treat the Coulomb interaction separately from the short-ranged interaction, as the Coulomb problem can be solved exactly in terms of Coulomb functions, which take on the role of the Hankel functions in this problem.

References

• Griffiths, J. D. (1995). Introduction to Quantum Mechanics. Pearson Prentice Hall. ISBN 0-13-111892-7.

External links

1. https://web.archive.org/web/20120425231714/http://homepages.rpi.edu/~napolj/Talks/PWALunch9Sep03.pdf
2. http://quantummechanics.ucsd.edu/ph130a/130_notes/node228.html