Quantum scattering

introduction

\(\varphi_{xx}+(\lambda-u)\varphi=0\)
discrete spectrum \(\lambda<0\)
continuous spectrum \(\lambda>0\)
for lists http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)

continuous spectrum

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

time independent Schrodinger equation

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

\(\varphi_{xx}+(\lambda-u(x))\varphi=0\)

delta potential example

\(V(x) = \lambda\delta(x)\)

\(\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = A_{\mathrm r}e^{ikx} + A_{\mathrm l}e^{-ikx}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) = B_{\mathrm r}e^{ikx} + B_{\mathrm l}e^{-ikx}, & \text{ if } x>0, \end{cases}\)

harmonic oscillator

harmonic oscillator in quantum mechanics

example

\[Lambda] := -1
u[x_] := -2 Sech[x]^2
f[x_] := Sech[x]
Simplify[D[D[f[x], x], x] + (\[Lambda] - u[x]) f[x]]
Plot[u[x], {x, -5, 5}]

history

http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

inverse scattering method

encyclopedia

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experts on the field

http://arxiv.org/