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Simple (?) solving of equations re Tunneling

  1. Dec 21, 2013 #1
    Revising for my exams, stuck on how to do this:

    http://imageshack.us/photo/my-images/594/mr15.jpg/

    please could someone tell me how to get (A/F)?

    thanks for any help!!

    it's not a homework question, just revision so a direct solution would be more than welcome.
     
  2. jcsd
  3. Dec 21, 2013 #2

    vanhees71

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    A bit more context would be good!
     
  4. Dec 21, 2013 #3
    Hi, not sure what you mean, they get A/F from those equations.

    If you mean where did they come from, it's an electron approaching a potential barrier and tunneling through it. there are 3 equations for the wavefunction, the incident electron (which has approaching and reflection componants), the tunelling wavefunction, which has a wavefunction with a potential componant (and also a reflective componant), and the exiting electron which has no reflective componant.

    approaching the barrier the soln of PSI is:

    PSI 1 = A exp ik1x + B exp -k1x

    inside the barrier the soln of PSI is:

    PSI 2 = C exp -k2x + D exp k2x

    where k = sqrt(2m(E-U)) / h-bar

    leaving the barrier

    PSI 3 = F exp ik1x

    at the boundary of the barrier PSI 1 = PSI 2 . thus differential of PSI 1 = differential of PSI 2

    at the other boundary of the barrier PSI 2 = PSI 3 . thus differential of PSI 2 = differential of PSI 3

    this is where the terms in the photo come from.

    know where the terms come from, just dont know how to solve for A / F
     
  5. Dec 24, 2013 #4
    ok this is more complicated than it looks lol

    an equally good derivation is for B^2 / A^2 which is the reflection coefficient, if A / F comes up in the exam im buggered, but i did figure it for B^2 / A^2

    here are the steps:

    do k2 X [1] + [2] to give [6]
    do k2 X [1] - [2] to give [7]

    sub [3] into [4] for F exp ik1L to give [8]

    put [6] and [7] for D and C into [8] to give [9]

    divide through by B, rearrange and take the modulus

    QED.
     
  6. Jan 1, 2014 #5
    derivation for A/F in detail here!!

    http://lynx.uio.no/trine/fys3510/alpha_transmission_eng.pdf [Broken]
     
    Last edited by a moderator: May 6, 2017
  7. Jan 1, 2014 #6

    vanhees71

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    Note that this is only an approximation for the Coulomb barrier. The Coulomb potential has no plane waves as asymptotic states!
     
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