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Double Delta Function Potential

  1. Oct 25, 2011 #1
    I have

    V (x) = [itex]\sqrt{((h-bar ^{2})V_{0})/2m}[/itex] [[itex]\delta(x-a)[/itex]+ [itex]\delta(x+a)[/itex]]

    How do I find R and T?

    Under what condition is there resonant transmission?
     
    Last edited: Oct 25, 2011
  2. jcsd
  3. Oct 25, 2011 #2

    Matterwave

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    How did you solve the 1 delta function potential? Do something similar here.
     
  4. Oct 25, 2011 #3
    well, Yes I have the solution for that but I don't understand it that much. Please help. Thanks
     
  5. Oct 25, 2011 #4

    Matterwave

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    So, for the scattering states, you can assume plane waves. You can, for example, have an incident plane wave from the left, and then you can have waves going in both directions in the middle and on the right. And then you would need to match boundary conditions.
     
  6. Oct 25, 2011 #5
    ok. I got these answers. Are these correct? Someone please tell me.

    General Equations

    [itex]U_{I}[/itex] = [itex]e^{ikx}[/itex] + R [itex]e^{-ikx}[/itex]


    [itex]U_{II}[/itex] = A [itex]e^{ikx}[/itex] + B [itex]e^{-ikx}[/itex]


    [itex]U_{III}[/itex] =T [itex]e^{-ikx}[/itex]


    Boundary Conditions

    if a = 0

    [itex]U_{I}[/itex] = [itex]U_{II}[/itex]

    1 + R = A + B

    [itex]U_{II}[/itex] = [itex]U_{III}[/itex]

    A + B = T



    discontinuity equation

    [itex]U'_{I}[/itex] - [itex]U'_{II}[/itex] = - [itex]\sqrt{\frac{2m V_{o}}{h-bar^{2}}}[/itex] [itex]U_{a}[/itex]

    ik (1 - R) - ik (A - B) = - [itex]\sqrt{\frac{2m V_{o}}{h-bar^{2}}}[/itex] R


    [itex]U'_{II}[/itex] - [itex]U'_{III}[/itex] = - [itex]\sqrt{\frac{2m V_{o}}{h-bar^{2}}}[/itex][itex]U_{a}[/itex]


    ik (A-B) - ikT = - [itex]\sqrt{\frac{2m V_{o}}{h-bar^{2}}}[/itex] T


    /// i hope someone can tell me if these are correct so I can continue my calculations. Thanks.
     
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