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.
The following description follows the canonical way of introducing elementary scattering theory.
A steady beam of particles scatters off a spherically symmetric potential
representing the particle beam should be solved: We make the following ansatz: where
that is of interest, because observations near the scattering center (e.g. an atomic nucleus) are mostly not feasible, and detection of particles takes place far away from the origin.
This suggests that it should have a similar form to a plane wave, omitting any physically meaningless parts.
We therefore investigate the plane-wave expansion: The spherical Bessel function
asymptotically behaves like This corresponds to an outgoing and an incoming spherical wave.
at large distances and set the asymptotic form of the scattered wave to where
is the so-called scattering amplitude, which is in this case only dependent on the elevation angle
In conclusion, this gives the following asymptotic expression for the entire wave function: In case of a spherically symmetric potential
, the scattering wave function may be expanded in spherical harmonics, which reduce to Legendre polynomials because of azimuthal symmetry (no dependence on
): 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.
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 asymptotic wave function is 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, k) is defined from it follows that and thus the differential cross section is given by This works for any short-ranged interaction.
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