14.1: Fundamentals of Scattering Theory
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider time-independent, energy conserving scattering in which the Hamiltonian of the system is written H=H0+V(r), where H0=p22m≡−ℏ22m∇2 is the Hamiltonian of a free particle of mass m, and V(r) the scattering potential. This potential is assumed to only be non-zero in a fairly localized region close to the origin. Let ψ0(r)=√neik⋅r represent an incident beam of particles, of number density n, and velocity v=ℏk/m. [See Equation ([e14.14gg]).] Of course, H0ψ0=Eψ0, where E=ℏ2k2/(2m) is the particle energy. Schrödinger’s equation for the scattering problem is (H0+V)ψ=Eψ, subject to the boundary condition ψ→ψ0 as V→0.
The previous equation can be rearranged to give (∇2+k2)ψ=2mℏ2Vψ. Now, (∇2+k2)u(r)=ρ(r) is known as the Helmholtz equation. The solution to this equation is well known u(r)=u0(r)−∫eik|r−r′|4π|r−r′|ρ(r′)d3r′. Here, u0(r) is any solution of (∇2+k2)u0=0. Hence, Equation ([e15.6]) can be inverted, subject to the boundary condition ψ→ψ0 as V→0, to give
ψ(r)=ψ0(r)−2mℏ2∫eik|r−r′|4π|r−r′|V(r′)ψ(r′)d3r′
Let us calculate the value of the wavefunction ψ(r) well outside the scattering region. Now, if r≫r′ then |r−r′|≃r−ˆr⋅r′ to first-order in r′/r, where ˆr/r is a unit vector that points from the scattering region to the observation point. It is helpful to define k′=kˆr. This is the wavevector for particles with the same energy as the incoming particles (i.e., k′=k) that propagate from the scattering region to the observation point. Equation ([e15.9]) reduces to
ψ(r)≃√n[eik⋅r+eikrrf(k,k′)], where
f(k,k′)=−m2π√nℏ2∫e−ik′⋅r′V(r′)ψ(r′)d3r′. The first term on the right-hand side of Equation ([e15.11]) represents the incident particle beam, whereas the second term represents an outgoing spherical wave of scattered particles.
The differential scattering cross-section, dσ/dΩ, is defined as the number of particles per unit time scattered into an element of solid angle dΩ, divided by the incident particle flux. From Section [s7.2], the probability flux (i.e., the particle flux) associated with a wavefunction ψ is j=ℏmIm(ψ∗∇ψ). Thus, the particle flux associated with the incident wavefunction ψ0 is
j=nv, where v=ℏk/m is the velocity of the incident particles. Likewise, the particle flux associated with the scattered wavefunction ψ−ψ0 is j′=n|f(k,k′)|2r2v′, where v′=ℏk′/m is the velocity of the scattered particles. Now, dσdΩdΩ=r2dΩ|j′||j|, which yields
dσdΩ=|f(k,k′)|2. Thus, |f(k,k′)|2 gives the differential cross-section for particles with incident velocity v=ℏk/m to be scattered such that their final velocities are directed into a range of solid angles dΩ about v′=ℏk′/m. Note that the scattering conserves energy, so that |v′|=|v| and |k′|=|k|.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)