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Schrodinger time independent equation for steps and barriers

  1. Jun 3, 2017 #1
    Ok for starters, im sorry for the lenght of the question and jamming multiple questions in here i feel like my doubts are all related and it would be best just to put it all in one place instead of creating 2-4 topics.
    I'm beginning to feel confused with the amount of ways i see people solving this.
    The book im following isnt helping much
    1. The problem statement, all variables and given/known data
    so im dividing this in two; a step and a barrier, both finite.
    I want to find the reflection and transmission coefficients of a step/barrier

    2. Relevant equations
    Ψ''(x)=-2m/ħ2(E-V)Ψ(x) for V=0 k2=E2m/ħ2 for V=/=0 (E-V)2m/ħ2=q2.


    3. The attempt at a solution
    a)
    so for the step we have two cases e<V and e>V, the step begins at x=0
    When e<V:
    for area x<0
    Ψ(x)= reikx+e-ikx ; Ψ'(x)= rikeikx-ike-ikx
    for x>0
    Ψ(x)= te-qx ; Ψ'(x)= -qte-qx
    This choice of coefficients is confusing to me (also see underlined part bellow), most of the material i see has A, B and C, but i remember using "r" and "t" in class, and since the negative exponential represent the wave moving to one direction and the positive ones represent the opposite direction it'd make sense to use this set up, but i feel the results i get are wrong.
    For example: r+1=t this doesn't really make sense does it?
    anyway, for "r" and "t" coeficients i get:
    t=(2ik)/(-q+ik) and T=(q/k)|t|2 so T= 4kq/(q2+k2)
    for A,B,C i get:
    C/A=(2ik)/(-q+ik) and T=(q/k)|C/A|2
    So this appears to be correct
    b)for e>V i get two complex exponential with coeficients A,B,C,D or 1,r,tD
    and i have no idea how to solve them

    c)
    now for the barrier of lenght L and e<V
    x<0
    Ψ(x)= Aeikx+Be-ikx ; Ψ'(x)= Aikeikx-Bike-ikx
    x>L
    Ψ(x)= Ceikx+De-ikx ; Ψ'(x)= Cikeikx-Dike-ikx
    and C must be 0 so it doesnt blow up to infinity.
    0<x<L
    e-V is negative so
    Ψ''(x)=-2m/ħ2(E-V)Ψ(x) becomes Ψ''(x)=2m/ħ2(V-e)Ψ(x)
    so
    Ψ(x)= Eeqx+Fe-qx ; Ψ'(x)= qEeqx-qEe-qx
    And now i set the negative exponentials to be the wave moving to the right (which i read it should be the positive exponentials moving to the right due to Euler's formula but there is no positive exponential so ????)
    T=|D/B|2
    and R = |A/B|2
    And so this is solvable... how?
    so i set the boundary cond. and got:
    Ψ(0)= A+Be=E+F (1.)
    Ψ'(0)= Aik-Bike=qE-qE (2.)
    De-ikL=EeqL+Fe-qL (3.)
    -Dike-ikL=qEeqL-qEe-qL (4.)
    I did (2)+ik(1), then (4)/(3) and got stuck here:
    [-(EeqL-Ee-qL)/(EeqL+Fe-qL)]*(A-B)=(E-F)
    im stuck.
    I have that feeling i have to understand something im missing, or there's a specific path im not taking and im just making it harder on me

    d) for e>V i feel it would be simmilar since the only difference is that all the exponentials are complex.
     
    Last edited: Jun 3, 2017
  2. jcsd
  3. Jun 4, 2017 #2
    Always make sure sure you have the correct number of equations for solving the system. In these problems, if you have n coefficients you would expect n-1 equations such that all the other coefficients can be expressed in terms of the incoming wave coefficient. These equations come from matching boundary conditions (Hint: Match them at every step/barrier, the ks change! I believe that's where you're missing equations). If you have enough eqtns, the rest is algebra.
     
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