Born Approximation

Quantum Mechanics · Part 8

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Table of Contents

  1. 1.The Scattering Setup
  2. 2.Lippmann-Schwinger Equation
  3. 3.The Free-Particle Green's Function
  4. 4.The First Born Approximation
  5. 5.Scattering Amplitude as Fourier Transform
  6. 6.Momentum Transfer
  7. 7.Spherically Symmetric Potentials
  8. 8.Iterative Structure of the Solution
  9. 9.Physical Picture

Key Equations

$$\psi(\vec{r}) \xrightarrow{r\to\infty} e^{i\vec{k}_i\cdot\vec{r}} + f(\theta,\phi)\,\frac{e^{ikr}}{r}$$
$$\left(\nabla^2 + k^2\right)G_0(\vec{r},\vec{r}\,') = \frac{2m}{\hbar^2}\,\delta^{(3)}(\vec{r}-\vec{r}\,')$$
$$|\vec{r} - \vec{r}\,'| \approx r - \hat{r}\cdot\vec{r}\,' \qquad \Rightarrow \qquad \frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|} \approx \frac{e^{ikr}}{r}\,e^{-i\vec{k}_f\cdot\vec{r}\,'}$$
$$|\vec{q}| = 2k\sin\!\left(\frac{\theta}{2}\right)$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the Quantum Mechanics course on CoursesHub.World. Master quantum mechanics from mathematical foundations to advanced topics. Covers Hilbert spaces, wave functions, angular momentum, perturbation theory, scattering, and path integrals with 450+ pages ...

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