14.2: Born Approximation
( \newcommand{\kernel}{\mathrm{null}\,}\)
Equation ([e15.17]) is not particularly useful, as it stands, because the quantity f(k,k′) depends on the, as yet, unknown wavefunction ψ(r). [See Equation ([e5.12]).] Suppose, however, that the scattering is not particularly strong. In this case, it is reasonable to suppose that the total wavefunction, ψ(r), does not differ substantially from the incident wavefunction, ψ0(r). Thus, we can obtain an expression for f(k,k′) by making the substitution ψ(r)→ψ0(r)=√nexp(ik⋅r) in Equation ([e5.12]). This procedure is called the Born approximation .
The Born approximation yields f(k,k′)≃m2πℏ2∫ei(k−k′)⋅r′V(r′)d3r′. Thus, f(k,k′) becomes proportional to the Fourier transform of the scattering potential V(r) with respect to the wavevector q=k−k′.
For a spherically symmetric potential, f(k′,k)≃−m2πℏ2∫∫∫exp(iqr′cosθ′)V(r′)r′2dr′sinθ′dθ′dϕ′, giving f(k′,k)≃−2mℏ2q∫∞0r′V(r′)sin(qr′)dr′. Note that f(k′,k) is just a function of q for a spherically symmetric potential. It is easily demonstrated that q≡|k−k′|=2ksin(θ/2), where θ is the angle subtended between the vectors k and k′. In other words, θ is the scattering angle. Recall that the vectors k and k′ have the same length, via energy conservation.
Consider scattering by a Yukawa potential ,
V(r)=V0exp(−μr)μr, where V0 is a constant, and 1/μ measures the “range” of the potential. It follows from Equation ([e17.38]) that f(θ)=−2mV0ℏ2μ1q2+μ2, because ∫∞0exp(−μr′)sin(qr′)dr′=qq2+μ2. Thus, in the Born approximation, the differential cross-section for scattering by a Yukawa potential is dσdΩ≃(2mV0ℏ2μ)21[2k2(1−cosθ)+μ2]2, given that q2=4k2sin2(θ/2)=2k2(1−cosθ).
The Yukawa potential reduces to the familiar Coulomb potential as μ→0, provided that V0/μ→ZZ′e2/(4πϵ0). In this limit, the Born differential cross-section becomes dσdΩ≃(2mZZ′e24πϵ0ℏ2)2116k4sin4(θ/2). Recall that ℏk is equivalent to |p|, so the previous equation can be rewritten dσdΩ≃(ZZ′e216πϵ0E)21sin4(θ/2), where E=p2/2m is the kinetic energy of the incident particles. Of course, Equation ([e17.46]) is identical to the famous Rutherford scattering cross-section formula of classical physics .
The Born approximation is valid provided that ψ(r) is not too different from ψ0(r) in the scattering region. It follows, from Equation ([e15.9]), that the condition for ψ(r)≃ψ0(r) in the vicinity of r=0 is |m2πℏ2∫exp(ikr′)r′V(r′)d3r′|≪1. Consider the special case of the Yukawa potential. At low energies, (i.e., k≪μ) we can replace exp(ikr′) by unity, giving 2mℏ2|V0|μ2≪1 as the condition for the validity of the Born approximation. The condition for the Yukawa potential to develop a bound state is 2mℏ2|V0|μ2≥2.7, where V0 is negative . Thus, if the potential is strong enough to form a bound state then the Born approximation is likely to break down. In the high-k limit, Equation ([e17.47]) yields 2mℏ2|V0|μk≪1. This inequality becomes progressively easier to satisfy as k increases, implying that the Born approximation is more accurate at high incident particle energies.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)