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

• 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
  \(A_r + A_l - B_r - B_l = 0\)
  \(-A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l)\)
  
• special case of scattering problem

 

 

 

harmonic oscillator

• harmonic oscillator in quantum mechanics

 

 

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}]

 

 

 

history

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

 

 

related items

• inverse scattering method

 

 

encyclopedia

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