Cheat sheets/Quantum mechanical scattering

Schrodinger equation
(Assume potential is radially symmetric) $$\left [ -\frac{\hbar^2}{2m}\nabla^2+V(r)\right ]\psi(r)=E\psi(r)$$

If we put
 * $$U=\frac{2m}{\hbar}V(r)$$

then
 * $$\left [\nabla^2+k^2-U(r)\right]\psi(r)=0$$

Recall
$$\nabla = \partial_r \hat{r} + \frac{1}{r}\partial_\theta \hat{\theta} + \frac{1}{r\sin\theta} \partial_\phi \hat{\phi} $$

One important result
$$\left(\nabla^2+k^2\right)\frac{e^{ikr}}{r}=-4\pi\delta^3(\vec{r})$$

Green's function
$$G(\vec{r})=\int\frac{d^3p}{(2\pi)^3}e^{-i\vec{p}\cdot\vec{r}}\frac{1}{k^2-p^2}=-\frac{1}{4\pi r}e^{ikr}$$

Simple energy relationships
$$E=\frac{|\vec{p}|^2}{2m} = \frac{\hbar^2}{2m}|\vec{k}|^2=\frac{1}{2}m|\vec{v}|^2$$

Wavefunction
$$\psi_{\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{r}}+f(k,\theta,\phi)\frac{e^{ikr}}{r}$$
 * Incoming: $$e^{i\vec{k}\cdot\vec{r}}$$
 * Scattered: $$\frac{e^{ikr}}{r}$$

Scattering Amplitude: f(k,φ,θ)

 * $$f(k,\phi,\theta)=-{1 \over 4\pi} \int e^{-ik'r}u(r)\psi_k(r)\,dr$$


 * $$f=-\frac{m}{2\pi\hbar^2}\left\langle \phi_{k'}|V|\psi_k\right\rangle$$


 * Plane Wave: $$\phi_{k'}(r)=e^{ik'r}$$

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

Total cross section
$$\int d\Omega\,\sigma(\Omega)=\int^{2\pi}_0 d\phi \int^\pi_0 \sin\theta d\theta |f(\theta,\phi)|^2$$