Step function potential scattering

introduction

• Schrodinger equation

\[E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\]

step potential 1:

• From Classical to Quantum Mechanics: An Introduction to the Formalism, Foundations and Applications http://goo.gl/VN3OG 184p

• Let the potential is given by

\[\nu = \begin{cases} V_{0} & \text{ if } 0<x<a; \\ 0& \text{ otherwise } \end{cases}\]

• solution\[ \begin{cases} \psi_L(x)= A_r e^{i k_0 x} + A_l e^{-i k_0x}\quad x<0 \\ \psi_C(x)= B_r e^{i k_1 x} + B_l e^{-i k_1x}\quad 0<x<a \\ \psi_R(x)= C_r e^{i k_0 x} + C_l e^{-i k_0x}\quad x>a \end{cases} \] where $k_0=\sqrt{2m E/\hbar^{2}}$ if $x<0$ or $x>a$ and $k_1=\sqrt{2m (E-V_0)/\hbar^{2}}$ if $0<x<a$
• we impose two conditions on the wave function
  • the wave function be continuous in the origin
  • the first derivative of the wave function be continuous in the origin
• the coefficient must satisfy\[ \begin{cases} A_r + A_l - B_r - B_l = 0 \\ ik_0(A_r - A_l) = ik_1(B_r - B_l) \\ B_re^{iak_1}+B_le^{-iak_1}=C_re^{iak_0}+C_le^{-iak_0} \\ ik_1(B_re^{iak_1}-B_le^{-iak_1})=ik_0(C_re^{iak_0}-C_le^{-iak_0}) \end{cases} \]

scattering

• special case of scattering problem \(A_r=1, A_l=r, C_r=t , C_l = 0\)
• wave function

\[ \begin{cases} \psi_L(x)= e^{i k_0 x} + r e^{-i k_0x}\quad x<0 \\ \psi_C(x)= B_r e^{i k_1 x} + B_l e^{-i k_1x}\quad 0<x<a \\ \psi_R(x)= t e^{i k_0 x} \quad x>a \end{cases} \]

• coefficient satisfy

\[ \begin{cases} 1 + r - B_r - B_l = 0 \\ ik_0 (1-r)= ik_1(B_r - B_l) \\ B_re^{iak_1}+B_le^{-iak_1}=t e^{iak_0} \\ ik_1(B_re^{iak_1}-B_le^{-iak_1})=ik_0(te^{iak_0}) \end{cases} \]

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

step potential 2 : not corrected

• 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

\[ \psi(x) = \begin{cases} \psi_{\mathrm L}(x) = A_{\mathrm r}e^{ik_{0}x} + A_{\mathrm l}e^{-ik_{0}x}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) = B_{\mathrm r}e^{ik_{1}x} + B_{\mathrm l}e^{-ik_{1}x}, & \text{ if } x>0, \end{cases} \]

• 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

\[ \begin{cases} A_r + A_l - B_r - B_l = 0 \\ -A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l) \end{cases} \]

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