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 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 V ( r ) {displaystyle V(r)} , which is short ranged so that for large distances r → ∞ {displaystyle r o infty } , the particles behave like free particles. In principle, any particle should be described by a wave packet but we describe the scattering of a plane wave traveling along the z-axis exp ⁡ ( i k z ) {displaystyle exp(ikz)} instead, because wave packets are expanded in terms of plane waves and this is mathematically simpler. Because the beam is switched on for times long compared to the time of interaction of the particles with the scattering potential, a steady state is assumed. This means that the stationary Schrödinger equation for the wave function Ψ ( r ) {displaystyle Psi (mathbf {r} )} representing the particle beam should be solved: We make the following ansatz: where Ψ 0 ( r ) ∝ exp ⁡ ( i k z ) {displaystyle Psi _{0}(mathbf {r} )propto exp(ikz)} is the incoming plane wave and Ψ s ( r ) {displaystyle Psi _{mathrm {s} }(mathbf {r} )} is a scattered part perturbing the original wave function.It is the asymptotic form of Ψ s ( r ) {displaystyle Psi _{mathrm {s} }(mathbf {r} )} 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. At large distances, the particles should behave like free particles and Ψ s ( r ) {displaystyle Psi _{mathrm {s} }(mathbf {r} )} should therefore be a solution to the free Schrödinger equation. 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 j ℓ ( k r ) {displaystyle j_{ell }(kr)} asymptotically behaves like This corresponds to an outgoing and an incoming spherical wave. For the scattered wave function, only outgoing parts are expected. We therefore expect Ψ s ( r ) ∝ exp ⁡ ( i k r ) / r {displaystyle Psi _{mathrm {s} }(mathbf {r} )propto exp(ikr)/r} at large distances and set the asymptotic form of the scattered wave to where f ( θ , k ) {displaystyle f( heta ,k)} is the so-called scattering amplitude, which is in this case only dependent on the elevation angle θ {displaystyle heta } and the energy.In conclusion, this gives the following asymptotic expression for the entire wave function: In case of a spherically symmetric potential V ( r ) = V ( r ) {displaystyle V(mathbf {r} )=V(r)} , the scattering wave function may be expanded in spherical harmonics which reduce to Legendre polynomials because of azimuthal symmetry (no dependence on ϕ {displaystyle phi } ): 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: