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14.2: Born Approximation

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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(ikr) in Equation ([e5.12]). This procedure is called the Born approximation .

The Born approximation yields f(k,k)m2π2ei(kk)rV(r)d3r. Thus, f(k,k) becomes proportional to the Fourier transform of the scattering potential V(r) with respect to the wavevector q=kk.

For a spherically symmetric potential, f(k,k)m2π2exp(iqrcosθ)V(r)r2drsinθdθdϕ, giving f(k,k)2m2q0rV(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|kk|=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)=2mV02μ1q2+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(2mV02μ)21[2k2(1cosθ)+μ2]2, given that q2=4k2sin2(θ/2)=2k2(1cosθ).

The Yukawa potential reduces to the familiar Coulomb potential as μ0, provided that V0/μZZe2/(4πϵ0). In this limit, the Born differential cross-section becomes dσdΩ(2mZZe24πϵ02)2116k4sin4(θ/2). Recall that k is equivalent to |p|, so the previous equation can be rewritten dσdΩ(ZZe216πϵ0E)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 |m2π2exp(ikr)rV(r)d3r|1. Consider the special case of the Yukawa potential. At low energies, (i.e., kμ) we can replace exp(ikr) by unity, giving 2m2|V0|μ21 as the condition for the validity of the Born approximation. The condition for the Yukawa potential to develop a bound state is 2m2|V0|μ22.7, 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 2m2|V0|μk1. 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)


This page titled 14.2: Born Approximation is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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