Born approximation

Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named after Max Born who proposed this approximation in the early days of quantum theory development.

It is the perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small compared to the incident field on the scatterer.

For example, the scattering of radio waves by a light styrofoam column can be approximated by assuming that each part of the plastic is polarized by the same electric field that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution.

Starting with a physical model based on the Schrodinger wave equation for scattering from a potential V the scattering amplitude, f, requires knowing the full scattering wavefunction ${\displaystyle \psi }$, $${\displaystyle f(\mathbf {k} _{f},\mathbf {k} _{i})=-{\frac {\mu }{2\pi \hbar ^{2}}}\int \psi _{f}^{*}V(\mathbf {r} )\psi _{i}d^{3}r}$$ In the Born approximation, the initial and final wavefunctions approximated plane waves: $${\displaystyle f(\mathbf {k} _{f},\mathbf {k} _{i})=-{\frac {\mu }{2\pi \hbar ^{2}}}\int e^{-i\mathbf {k} _{f}\cdot \mathbf {r} }V(\mathbf {r} )e^{i\mathbf {k} _{i}\cdot \mathbf {r} }d^{3}r}$$ This is mathematically equivalent to the Fourier transform of the scattering potential from ${\displaystyle \mathbf {r} }$ to ${\displaystyle \mathbf {q} =\mathbf {k} _{f}-\mathbf {k} _{i}}$: $${\displaystyle f(\mathbf {k} _{f},\mathbf {k} _{i})=-{\frac {\mu }{2\pi \hbar ^{2}}}\int V(\mathbf {r} )e^{i\mathbf {q} \cdot \mathbf {r} }d^{3}r}$$ For a spherically symmetric potential the angular integrations can be performed and the scattering amplitude depends only on the polar angle ${\displaystyle \theta }$ between the input and output directions: $${\displaystyle f(\theta )=-{\frac {2\mu }{\hbar ^{2}q}}\int _{0}^{\infty }r\sin(qr)V(r)dr}$$ where ${\displaystyle q=2k\sin(\theta /2)}$ and ${\displaystyle k=|\mathbf {k} _{f}-\mathbf {k} _{i}|.}$

The Lippmann–Schwinger equation for the scattering state ${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }$ with a momentum p and out-going (+) or in-going (−) boundary conditions is

where ${\displaystyle G^{\circ }}$ is the free particle Green's function, ${\displaystyle \epsilon }$ is a positive infinitesimal quantity, and ${\displaystyle V}$ the interaction potential. ${\displaystyle \vert {\Psi _{\mathbf {p} }^{\circ }}\rangle }$ is the corresponding free scattering solution sometimes called the incident field. The factor ${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }$ on the right hand side is sometimes called the driving field.

The Born approximation sets $${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle \approx \vert {\Psi _{\mathbf {p} }^{\circ }}\rangle }$$ Within the Born approximation, the above equation is expressed as

which is much easier to solve since the right hand side no longer depends on the unknown state ${\displaystyle \vert {\Psi _{\mathbf {p} }^{(\pm )}}\rangle }$.