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Coeficients of a time independent schrodinger equation approaching a finite potential

  1. Nov 18, 2009 #1
    1. The problem statement, all variables and given/known data
    I am trying to find the coefficients in a Schrodinger equation approaching a finite potential.
    https://www.physicsforums.com/showthread.php?t=203385
    It is a problem similar to this, except a little easier. In my case, though, there is no V1 as shown in the picture at the thread, the potential is at x = 0 and doesn't drop back down, just continues going in the positive x direction. So at x = 0 it just goes straight up vertically, then levels off and keeps going, creating a sort of wall. (hope that is a good enough explanation)

    2. Relevant equations
    Schrodingers Equation


    3. The attempt at a solution
    I have defined the regions as:
    Region 1: V(x) = 0 for x < 0
    Region 2: V(x) = V0 for x > 0

    I have used Schrodinger's Equation to get the wave function for each region:
    Region 1:
    Psi(x) = Aeik1x + Be-ik1x
    With A being the incident wave and B being the reflected wave off of the potential barrier.
    I also have K1 = (2mE/h2)1/2

    Region 2:
    Psi(x) = Ceik2x
    With C being for the continuing wave, there is no reflected wave here because region 2 contains the barrier. I also think C would be considered T (transmitted).
    I also have K2 = (2m(E-V0)/h2)1/2

    So I can say that A + B = C at x = 0 (I think). I can also say that their derivatives are equal at that point, so ik1A - ik2B = ik2C

    I also know that 1 = P(R) + P(T)
    and R = abs(B2/A2)
    and I have written T = 1 - R

    I guess I'm supposed to solve B and C in terms of A, I'm almost positive all of the information thus far is correct, unless I just typed something incorrectly.

    It's something that I haven't ever seen/done before so I'm absolutely stumped... I even looked at some old physics books from the library trying to get some ideas but am lost. I'm sure it's something really simple... I just need to know where to go from this point in order to solve for A B and C.

    Thanks!
     
    Last edited: Nov 18, 2009
  2. jcsd
  3. Nov 18, 2009 #2

    lanedance

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    Homework Helper

    Re: Coeficients of a time independent schrodinger equation approaching a finite poten

    so you have 2 equations linking A,B & C due to the boundary conditions (continuous wave function & derivative)

    you should be able to solve for B & C in terms of A...
     
  4. Nov 18, 2009 #3
    Re: Coeficients of a time independent schrodinger equation approaching a finite poten

    Well, it's not too bad. You can relate everything using your two equations at the boundary, and finding an expression for C/A and B/A. These expressions should end up containing k1 and k2.

    You can then find a way to relate C/A and B/A to transmission and reflection coefficients.

    I'm pretty sure most quantum texts discuss this - try ch. 7 of Liboff's book, or ch. 5 of Shankar's. This would be best classified as the single step potential.
     
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