Quantum scattering

introduction

Schrodinger equation
\(E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\)
simplified form
\(-\varphi_{xx}+u(x)\varphi = \lambda\varphi\)
\(\varphi_{xx}+(\lambda-u(x))\varphi=0\)

e^{ikx} represents a wave traveling to the right, and e^{−ikx} one traveling to the left
e^{−ikx} is incoming wave from the right to the left
reflection and transmission coefficient
\(\varphi \sim e^{-ikx}+\rho(k,t)e^{ikx}\) as \(x\to +\infty\)
\(\varphi \sim \tau(k,t)e^{-ikx}\) as \(x\to -\infty\)
\(\rho(k,t)\) and \(\tau(k,t)\) are called the reflection and transmission coefficient

\(r=t-1\)

If t is of the form \(t=\frac{1}{1-ai}\) (real number a), then

\(|r|^2+|t|^2=1\)