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Physics LibreTexts

14.1: Fundamentals of Scattering Theory

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Consider time-independent, energy conserving scattering in which the Hamiltonian of the system is written H=H0+V(r), where H0=p22m22m2 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)=neikr represent an incident beam of particles, of number density n, and velocity v=k/m. [See Equation ([e14.14gg]).] Of course, H0ψ0=Eψ0, where E=2k2/(2m) is the particle energy. Schrödinger’s equation for the scattering problem is (H0+V)ψ=Eψ, subject to the boundary condition ψψ0 as V0.

The previous equation can be rearranged to give (2+k2)ψ=2m2Vψ. 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|rr|4π|rr|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 ψψ0 as V0, to give

ψ(r)=ψ0(r)2m2eik|rr|4π|rr|V(r)ψ(r)d3r

Let us calculate the value of the wavefunction ψ(r) well outside the scattering region. Now, if rr then |rr|rˆrr 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[eikr+eikrrf(k,k)], where

f(k,k)=m2πn2eikrV(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=mIm(ψψ). Thus, the particle flux associated with the incident wavefunction ψ0 is

j=nv, where v=k/m is the velocity of the incident particles. Likewise, the particle flux associated with the scattered wavefunction ψψ0 is j=n|f(k,k)|2r2v, where v=k/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=k/m to be scattered such that their final velocities are directed into a range of solid angles dΩ about v=k/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)


This page titled 14.1: Fundamentals of Scattering Theory is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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