Part VIII, Chapter 2 | Page 1 of 3

The Born Approximation

Perturbative approach to scattering: weak potentials treated as perturbations on plane waves

The Scattering Setup

The fundamental scattering problem considers a beam of particles incident on a localized potential $V(\vec{r})$. Far from the scattering center the total wavefunction must take the form of an incident plane wave plus an outgoing scattered spherical wave:

$$\psi(\vec{r}) \xrightarrow{r\to\infty} e^{i\vec{k}_i\cdot\vec{r}} + f(\theta,\phi)\,\frac{e^{ikr}}{r}$$

Here $e^{i\vec{k}_i\cdot\vec{r}}$ represents the incident plane wave with wave vector $\vec{k}_i$, while $f(\theta,\phi)$ is the scattering amplitude that encodes all angular information about the scattering process. The differential cross section is directly given by:

$$\frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2$$

The central challenge is to determine $f(\theta,\phi)$ for a given potential. The Born approximation provides an elegant perturbative solution when the potential is weak compared to the kinetic energy.

Lippmann-Schwinger Equation

The time-independent Schrodinger equation for scattering,$(H_0 + V)|\psi\rangle = E|\psi\rangle$, can be recast into an integral equation. Let $|\phi\rangle$ be the free-particle state satisfying $H_0|\phi\rangle = E|\phi\rangle$. The full scattering state obeys:

$$|\psi\rangle = |\phi\rangle + G_0(E)\,V\,|\psi\rangle$$

This is the Lippmann-Schwinger equation. Here $G_0(E)$ is the free-particle Green's operator (resolvent):

$$G_0(E) = \frac{1}{E - H_0 + i\epsilon} = \lim_{\epsilon\to 0^+}\frac{1}{E - H_0 + i\epsilon}$$

The $+i\epsilon$ prescription selects outgoing spherical waves (retarded boundary conditions). The equation is exact but implicit—the unknown $|\psi\rangle$ appears on both sides.

The Free-Particle Green's Function

In position representation, the Green's operator becomes the Green's function. For a particle of mass $m$ with energy $E = \hbar^2 k^2/(2m)$, it satisfies:

$$\left(\nabla^2 + k^2\right)G_0(\vec{r},\vec{r}\,') = \frac{2m}{\hbar^2}\,\delta^{(3)}(\vec{r}-\vec{r}\,')$$

The outgoing-wave solution is:

$$G_0(\vec{r},\vec{r}\,') = -\frac{1}{4\pi}\frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|}$$

This can be derived by Fourier transform or by recognizing that $e^{ikR}/R$(where $R = |\vec{r}-\vec{r}\,'|$) solves the Helmholtz equation for $R \neq 0$. Inserting this Green's function, the Lippmann-Schwinger equation becomes:

$$\psi(\vec{r}) = e^{i\vec{k}_i\cdot\vec{r}} - \frac{m}{2\pi\hbar^2}\int \frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|}\,V(\vec{r}\,')\,\psi(\vec{r}\,')\ d^3r'$$

The First Born Approximation

The key idea: if $V$ is weak, the wavefunction $\psi(\vec{r}\,')$inside the integral barely differs from the incident plane wave. The first Born approximation replaces $\psi(\vec{r}\,') \approx e^{i\vec{k}_i\cdot\vec{r}\,'}$:

$$\psi^{(1)}(\vec{r}) = e^{i\vec{k}_i\cdot\vec{r}} - \frac{m}{2\pi\hbar^2}\int \frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|}\,V(\vec{r}\,')\,e^{i\vec{k}_i\cdot\vec{r}\,'}\ d^3r'$$

In the far field ($r \gg r'$), we use the asymptotic expansion:

$$|\vec{r} - \vec{r}\,'| \approx r - \hat{r}\cdot\vec{r}\,' \qquad \Rightarrow \qquad \frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|} \approx \frac{e^{ikr}}{r}\,e^{-i\vec{k}_f\cdot\vec{r}\,'}$$

where $\vec{k}_f = k\hat{r}$ is the final wave vector pointing toward the detector. Comparing with the asymptotic form, we extract the scattering amplitude:

First Born Scattering Amplitude

$$\boxed{f^{(1)}(\theta,\phi) = -\frac{m}{2\pi\hbar^2}\int e^{-i\vec{q}\cdot\vec{r}\,'}\,V(\vec{r}\,')\ d^3r'}$$

Scattering Amplitude as Fourier Transform

The Born result has a beautiful interpretation: the scattering amplitude is proportional to the three-dimensional Fourier transform of the potential, evaluated at the momentum transfer:

$$f^{(1)}(\vec{q}) = -\frac{m}{2\pi\hbar^2}\,\widetilde{V}(\vec{q})$$

Momentum Transfer

The momentum transfer vector $\vec{q}$ is defined as the difference between the incident and final wave vectors:

$$\vec{q} = \vec{k}_f - \vec{k}_i$$

For elastic scattering, $|\vec{k}_i| = |\vec{k}_f| = k$, and the magnitude of the momentum transfer depends only on the scattering angle $\theta$:

$$|\vec{q}| = 2k\sin\!\left(\frac{\theta}{2}\right)$$

This follows from the law of cosines: $q^2 = k^2 + k^2 - 2k^2\cos\theta = 2k^2(1-\cos\theta) = 4k^2\sin^2(\theta/2)$.

Physical Insight

Small $q$ (forward scattering, small $\theta$): probes long-range structure of potential
Large $q$ (backward scattering, large $\theta$): probes short-range structure of potential
Scattering experiments effectively "measure" the Fourier components of the potential

Spherically Symmetric Potentials

When the potential depends only on the radial distance, $V = V(r)$, the angular integration can be performed analytically. Choosing the $\vec{q}$ axis as the polar axis and performing the $\phi'$ and $\theta'$ integrations:

$$f(q) = -\frac{2m}{\hbar^2 q}\int_0^\infty V(r')\,r'\sin(qr')\,dr'$$

For a spherical potential the scattering amplitude depends only on $q = 2k\sin(\theta/2)$, and hence only on the scattering angle. The azimuthal symmetry means $f$ is independent of $\phi$. The differential cross section is:

$$\frac{d\sigma}{d\Omega} = |f(q)|^2 = \left|\frac{2m}{\hbar^2 q}\int_0^\infty V(r')\,r'\sin(qr')\,dr'\right|^2$$

This one-dimensional integral is often analytically tractable, making the Born approximation a powerful computational tool.

Iterative Structure of the Solution

The Lippmann-Schwinger equation has a beautiful iterative structure. We can solve it by successive substitution:

The $n$-th term represents $n$-fold scattering from the potential. The first Born approximation retains only the first correction term $G_0 V|\phi\rangle$, corresponding to single scattering. Each subsequent term adds another interaction with the potential, with free propagation ($G_0$) between interactions.

This perturbative expansion converges when the operator norm $\|G_0 V\| < 1$, which is essentially the Born validity condition. When it converges, even the first term often gives an excellent approximation at high energies.

Physical Picture

Summary of the Born Approach

Input: the potential $V(\vec{r})$ and incident energy$E = \hbar^2 k^2/(2m)$
Approximation: replace $\psi \to \phi$ (plane wave) inside the scattering integral
Result: scattering amplitude is the 3D Fourier transform of the potential evaluated at the momentum transfer $\vec{q} = \vec{k}_f - \vec{k}_i$
Limitation: valid only when $V_0 \ll \hbar^2/(2ma^2)$(weak potential) or $E \gg V_0$ (high energy)
Connection to experiment: measuring $d\sigma/d\Omega$ at different angles maps out the Fourier transform of the potential

The Born approximation is the workhorse of high-energy scattering theory. In the next page, we apply it to specific potentials and derive explicit cross sections.

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