Step function potential scattering

introduction

• Let the potential is given by
  \(\psi(x) = \begin{cases} V_{-} & \text{ if } x<0; \\ V_{+}, & \text{ if } x>0, \end{cases}\)
  

• solution of the stationary S
  \(\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = A_{\mathrm r}e^{ik_{0}x} + A_{\mathrm l}e^{-ik_{0}x}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) = B_{\mathrm r}e^{ik_{1}x} + B_{\mathrm l}e^{-ik_{1x}, & \text{ if } x>0, \end{cases}\)
  
• 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
  \(\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = e^{ikx} + re^{-ikx}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) =te^{ikx} , & \text{ if } x>0, \end{cases}\)
  

 

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

 

 

 

history

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

 

 

 

 

encyclopedia

• http://en.wikipedia.org/wiki/Rectangular_potential_barrier
• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://eom.springer.de
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2011년 books and articles
• http://library.nu/search?q=
• http://library.nu/search?q=

 

 

 

 

 

articles

 

• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/