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.
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Preliminary scattering theory
The following description follows the canonical way of introducing elementary scattering theory. A steady beam of particles scatters off a spherically symmetric potential
We make the following ansatz:
where
The spherical Bessel function
This corresponds to an outgoing and an incoming spherical wave. For the scattered wave function, only outgoing parts are expected. We therefore expect
where
Partial wave expansion
In case of a spherically symmetric potential
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 of spherical Bessel functions and Legendre polynomials:
Here, we have assumed a spherical coordinate system in which the z-axis is aligned with the beam direction. The radial part of this wave function consists solely of the spherical Bessel function, which can be rewritten as a sum of two spherical Hankel functions:
This has physical significance: hℓ(2) asymptotically (i.e. for large r) behaves as i−(ℓ+1)eikr/(kr) and is thus an outgoing wave, whereas hℓ(1) asymptotically behaves as iℓ+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 Sℓ:
where uℓ(r)/r is the radial component of the actual wave function. The scattering phase shift δℓ is defined as half of the phase of Sℓ:
If flux is not lost, then |Sℓ| = 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,
Subtracting ψin yields the asymptotic outgoing wave function:
Making use of the asymptotic behavior of the spherical Hankel functions, one obtains:
Since the scattering amplitude f(θ, φ) is defined via:
It follows that
and thus the differential cross section is given by
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.