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)

 

 

time independent Schrodinger equation

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

 

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
  

 

potential scattering

\(r=t-1\)

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

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

 

 

delta potential example

• delta potential scattering

 

 

harmonic oscillator

• harmonic oscillator in quantum mechanics

 

 

sech potential example

• sech potential example

 

1. \[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}]

 

 

 

related items

• inverse scattering method

 

 

encyclopedia

• http://en.wikipedia.org/wiki/Schrödinger_equation
  
• http://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
  
• http://en.wikipedia.org/wiki/Rectangular_potential_barrier
• http://en.wikipedia.org/wiki/Step_potential