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.

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 Schrödinger wave equation for scattering from a potential V , calculating the scattering amplitude, f requires knowing the full scattering wave function \psi , f(\mathbf{k}_f,\mathbf{k}_i) = -\frac{\mu}{2\pi\hbar^2} \int \psi_f^* \, V(\mathbf{r}) \, \psi_i \, d^3r In the Born approximation, the initial and final wave functions are approximated as plane waves: 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^3r This is mathematically equivalent to the Fourier transform of the scattering potential from \mathbf{r} to \mathbf{q}=\mathbf{k}_f - \mathbf{k}_i : 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^3r For a spherically symmetric potential the angular integrations can be performed and the scattering amplitude depends only on the polar angle \theta between the input and output directions: f(\theta) = -\frac{2\mu}{\hbar^2q} \int_0^\infty r \sin(qr) \, V(r) \,dr where q=2k \sin (\theta/2) and k=|\mathbf{k}_f - \mathbf{k}_i|. ==For the Lippmann–Schwinger equation== The Lippmann–Schwinger equation for the scattering state \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle with a momentum \mathbf{p} and out-going (+) or in-going (−) boundary conditions is \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle = \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle + G^\circ(E_p \pm i\epsilon) V \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle, where G^\circ is the free particle Green's function, \epsilon is a positive infinitesimal quantity, and V the interaction potential. \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle is the corresponding free scattering solution sometimes called the incident field. The factor \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle on the right hand side is sometimes called the driving field. The Born approximation sets \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle \approx \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle Within the Born approximation, the above equation is expressed as \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle = \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle + G^\circ(E_p \pm i\epsilon) V \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle, which is much easier to solve since the right hand side no longer depends on the unknown state \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle . The obtained solution is the starting point of a perturbation series known as the Born series. ===Scattering amplitude=== Using the outgoing free Green's function for a particle with mass m in coordinate space, G^{(+)}(\mathbf r, \mathbf r')=-\frac{2m}{\hbar^2}\frac{e^{+ik|\mathbf r-\mathbf r'|}}{4\pi|\mathbf r-\mathbf r'|} one can extract the Born approximation to the scattering amplitude from the Born approximation to the Lippmann–Schwinger equation above, f_B(\theta)=-\frac{m}{2\pi\hbar^2} \int e^{-i\mathbf{q} \cdot \mathbf{r}} \, V(\mathbf r) \,d^3r, where \theta is the angle between the incident wave vector \mathbf k and the scattered wave vector \mathbf k' , \mathbf q=\mathbf k'-\mathbf k is the transferred momentum. The Born scattering amplitude is proportional to the Fourier transform of the potential. In the centrally symmetric potential V=V(r) , the scattering amplitude becomes f_B(\theta) = -\frac{2m}{\hbar^2} \int_0^\infty r \, V(r) \, \frac{\sin (qr)}{q} \, dr where q=|\mathbf q|=2k\sin(\theta/2). In the Born approximation for centrally symmetric field, the scattering amplitude and thus the cross section \sigma depends on the momentum p=k/\hbar and the scattering amplitude \theta only through the combination p\sin(\theta/2) . In the distorted-wave Born approximation (DWBA), the incident waves are solutions \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle to a part V^1 of the problem V=V^1 + V^2 that is treated by some other method, either analytical or numerical. The interaction of interest V is treated as a perturbation V^2 to some system V^1 that can be solved by some other method. For nuclear reactions, numerical optical model waves are used. For scattering of charged particles by charged particles, analytic solutions for coulomb scattering are used. This gives the non-Born preliminary equation \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle = \vert{\Psi_{\mathbf{p}}^{\circ}}\rangle + G^\circ(E_p \pm i0) V^{1} \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle and the Born approximation \vert{\Psi_{\mathbf{p}}^{(\pm)}}\rangle = \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle + G^1(E_p \pm i0) V^{2} \vert{\Psi_{\mathbf{p}}^{1}}^{(\pm)}\rangle. Applications include bremsstrahlung and the photoelectric effect. For a charged-particle-induced direct nuclear reaction, the procedure is used twice. In condensed-matter research, DWBA is used to analyze grazing-incidence small-angle scattering. ==References== ==Further reading== Wu and Ohmura, Quantum Theory of Scattering, Prentice Hall, 1962

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