Delta potential scattering

introduction

• Let the potential is given by \(V(x) = \lambda\delta(x)\)
  \(\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = A_{\mathrm r}e^{ikx} + A_{\mathrm l}e^{-ikx}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) = B_{\mathrm r}e^{ikx} + B_{\mathrm l}e^{-ikx}, & \text{ if } x>0, \end{cases}\)
  
• we impose two conditions on the wave function
  
  •  the wave function be continuous in the origin
  •  integrate the Schrödinger equation around x = 0, over an interval [−ε, +ε] and In the limit as ε → 0, the right-hand side of this equation vanishes; the left-hand side becomes
• first condition
  \(\psi(0) =\psi_L(0) = \psi_R(0) = A_r + A_l = B_r + B_l\)
  \(A_r + A_l - B_r - B_l = 0\)
  
• second condition
  \( -\frac{\hbar^2}{2 m} \int_{-\epsilon}^{+\epsilon} \psi''(x) \,dx + \int_{-\epsilon}^{+\epsilon} V(x)\psi(x) \,dx = E \int_{-\epsilon}^{+\epsilon} \psi(x) \,dx\)
  LHS becomes \(-\frac{\hbar^2}{2m}[\psi_R'(0)-\psi_L'(0)] +\lambda\psi(0)\)
  RHS becomes 0
  \(-A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l)\)
  
• the coefficient must satisfy
  \(A_r + A_l - B_r - B_l = 0\)
  \(-A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l)\)
  

 

 

delta potential scattering

• special case of scattering problem \(A_r=1, A_l=r, B_r=t , B_l = 0\)
• wave function
  \(\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = e^{ikx} + re^{-ikx}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) =te^{ikx} , & \text{ if } x>0, \end{cases}\)
  

 

• \(t-r=1\)
  \(t=\cfrac{1}{1-\cfrac{m\lambda}{i\hbar^2k}}\,\!\)
  \(r=\cfrac{1}{\cfrac{i\hbar^2 k}{m\lambda} - 1}\,\!\)
  \(R=|r|^2=\cfrac{1}{1+\cfrac{\hbar^4k^2}{m^2\lambda^2}}= \cfrac{1}{1+\cfrac{2\hbar^2 E}{m\lambda^2}}.\,\!\)
  \(T=|t|^2=1-R=\cfrac{1}{1+\cfrac{m^2\lambda^2}{\hbar^4k^2}}= \cfrac{1}{1+\cfrac{m \lambda^2}{2\hbar^2 E}}\,\!\)
  

 

 

 

encyclopedia

• http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)