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Electron beam, potential barrier

  1. Apr 24, 2010 #1
    Electron beam with kinetic energy [tex] E_{k} = 10 eV[/tex] strikes a positive potential barrier [tex] V_{0}[/tex] and the kinetic energy after the beam has passed through the barrier is [tex] E_{k} = (10 eV -V_{0})[/tex].
    How big potential [tex] V_{0}[/tex] is needed so that 40% of the electron beam is going to be reflected?
    What would happen if we now make the potential negative so the electron beam will gain the kinetic energy?



    I would say that the energy of the particles is higher than the energy of the potential barrier, thats why we observe transmission and reflection
    solving The Schrödinger equation

    [tex] \frac{\partial^2}{\partial x^2}\psi(x)+\frac{2m}{\hbar^2}[E-V(x)]\psi(x)=0[/tex]
    then the solutions will be

    [tex] \psi_{1}=Ae^{ik_{1}x}+Be^{-ik_{1}x}[/tex] in the region x<0 [tex] k_{1}= \sqrt{\frac{2mE}{\hbar^2}}[/tex]
    [tex]\psi_{1}=Ce^{ik{2}x}[/tex] in the region x>0 [tex]k_{2}= \sqrt{\frac{2m[E-V_{0}]}{\hbar^2}}[/tex]

    the reflection coefficient is [tex] R= (\frac{k_{1}-k_{2}}{k_{1}+k_{2}})^2[/tex]

    Can someone help me with the solution?
    Is the reflection coefficient going to be 0.4? How to find that value of [tex] V_{0}[/tex][/QUOTE]

    I have calculted [tex] k_{1}= \sqrt{\frac{9.1\cdot10^{-31}\cdot10}{(6.582\cdot10^{-16})^{2}}[/tex]=[tex] 6.48[/tex]

    [tex]k_{1}= \sqrt{\frac{2mE}{\hbar^2}}= \sqrt{\frac{kg\cdot J}{J^2\cdot s^2}[/tex][tex]=\sqrt{\frac{kg}{N^2\cdot s^2}[/tex]=[tex] \frac{1}{m}[/tex]

    then i am trying to solve the equation with R

    [tex] R= (\frac{k_{1}-k_{2}}{k_{1}+k_{2}})^2[/tex] to calculate [tex] k_{2} [/tex] and finally calculate [tex] V_{0}[/tex] from the equation for [tex] k_{2} [/tex]

    but i am stuck here....

    [tex] R= \frac{k_{1}^2-2k_{1}k_{2}+k_{2}^2}{k_{1}^2+2k_{1}k_{2}+k_{2}^2}[/tex]
    and then i substitute for R= 04 and for [tex] k_{1}= 6.48[/tex] does anyone know i am correct so far?
    if yes, how would you solve this ?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Apr 24, 2010
  2. jcsd
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