The Born approximation

5.2 The Born approximation

Summary: If the potential is weak, we can use ﬁrst order perturbation theory to calculate cross sections.

If the scattering potential $V (r)$ is weak, we can ignore multiple interactions and use ﬁrst-order perturbation theory. In this context, this is called the Born approximation.

First order time-dependent perturbation theory means using Fermi’s golden rule. $V (r)$ is constant, not oscillatory, so the energy-conservation $δ$ -function links incoming and scattering states with the same energy, hence we are dealing with elastic scattering (as already assumed). Our goal is to calculate $d σ ∕ d Ω$ , so we are interested in all the out-going momentum states which fall within $d Ω$ at a given scattering angle ${θ, ϕ}$ :

\begin{array}{rcl} R_{k_{i} \to d Ω} & = & \sum_{k_{f} \in d Ω} \frac{2 π}{ℏ} {|⟨ k_{f} | V (r) | k_{i} ⟩|}^{2} δ (E_{i} - E_{f}) \\ = & \frac{2 π}{ℏ} \int_{0}^{\infty} {|⟨ k_{f} | V (r) | k_{i} ⟩|}^{2} δ (\frac{ℏ^{2}}{2 m} (k_{f}^{2} - k_{i}^{2})) D (k_{f}) d Ω d k_{f} \\ = & \frac{m V}{4 π^{2} ℏ^{3}} k_{f} {|⟨ k_{f} | V (r) | k_{i} ⟩|}^{2} d Ω \\ \Rightarrow d σ & = & \frac{m^{2} V^{2}}{4 π^{2} ℏ^{4}} {|⟨ k_{f} | V (r) | k_{i} ⟩|}^{2} d Ω \\ \Rightarrow |f (k, θ, ϕ)| & = & \frac{m}{2 π ℏ^{2}} \int e^{i (k_{i} - k_{f}) \cdot r} V (r) d^{3} r \end{array}

(Since the density of states is for single particles, we adjusted the normalisation of the incoming beam to also contain one particle in volume $V$ ; hence $V$ drops out. For the density of states, see A.10. For transforming the argument of delta functions, see A.8.)

Writing $k_{i} - k_{f} = q$ , the momentum transfered from the initial to the ﬁnal state, we have the very general result that $f (k, θ, ϕ)$ is just proportional to $Ṽ (q)$ , the Fourier transform of $V (r)$ .

The classic application is Rutherford scattering, but we will start with a Yukawa potential $V (r) = - λ e^{- μ r} ∕ r$ ; the Coulomb potential is the $μ \to 0$ limit. Taking the $z$ -axis along $q$ for the spatial integration (this is quite independent of the angles used in deﬁning the scattering direction) and noting that $q = 2 k sin (θ ∕ 2)$ , we get

\begin{array}{rcl} |f (k, θ, ϕ)| & = & \frac{m}{2 π ℏ^{2}} \int e^{i q r^{'} cos θ^{'}} λ e^{- μ r^{'}} r^{'} sin θ^{'} d θ^{'} d ϕ^{'} d r^{'} \\ = & \frac{2 m λ}{ℏ^{2} (μ^{2} + q^{2})} \\ \Rightarrow \frac{d σ}{d Ω} & = & \frac{4 m^{2} λ^{2}}{ℏ^{4} {(μ^{2} + 4 k^{2} {sin}^{2} (θ ∕ 2))}^{2}} \\ \Rightarrow {\frac{d σ}{d Ω}}_{Coulomb} & = & \frac{ℏ^{2} c^{2} α^{2}}{16 E^{2} {sin}^{4} (θ ∕ 2)} \end{array}

We note that the independence of $ϕ$ is quite general for a spherical potential.

This result, though derived at ﬁrst-order, is in fact correct to all orders and agrees with the classical expression, which is called the Rutherford cross section. (Just as well for Rutherford!) (The apearance of $ℏ$ is only because we have written $e^{2}$ in terms of $α$ .) The reason we took the limit of a Yukawa is that our formalism doesn’t apply to a Coulomb potential, because there is no asymptotic region - the potential is inﬁnite ranged. The cross section blows up at $θ = 0$ , the forward direction, but obviously we can’t put a detector there as it would be swamped by the beam.

Gasiorowicz ch 19.3
(Mandl ch 11.3)
Shankar ch 19.3
(Townsend ch 13.2)

(The references in brackets do not use the FGR to obtain the Born cross section.)

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