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Penetrating the Rectangular Potential Barrier, E < V0

  1. Jan 23, 2016 #1
    1. The problem statement, all variables and given/known data
    Rectangular barrier of width a and height V0 with an E < V0. What is the transmission ratio, T?

    2. Relevant equations
    T = C*C/A*A
    What am I missing in my attempted solution (variable name = TChip), which is different from the known solution (variable name = TSoln)?

    3. The attempt at a solution
    (shown in the Octave code below...note the actual solution has a negative concavity, whereas my solution has a positive concavity, but seems to be in the vicinity of the correct solution...this can be seen by running the Octave code).

    ****

    %I wrote this to check my work thus far on solving the GIANT algebra problem of penetrating the rectangular barrier, where E < V0
    clear;close;clc;
    % setting some constants

    hbar = 1.0545718*10^(-34);
    m = 9.10938356 * 10^(-31);
    a = 10^-20; %cab be purely arbitrary

    V0 = 100; %cab be purely arbitrary
    inc = 10;
    E = inc:inc:(V0-inc);

    %wavenumber formulas
    k1 = (sqrt(2*m*E))/hbar;
    k2 = (sqrt(2*m*(V0-E)))/hbar;

    %this section contains the solution
    TSoln = (1 + (((sinh(k2*a)).^2)./((4*E/V0).*(1-E/V0)))).^(-1)

    %this section contains my work thus far

    iVal = i*k2/k1;
    BLABLA1 = ( 1 - ((1-iVal).*(e.^(-2*k2*a) )./(1+iVal) ) ) + iVal.* (1 + ((1-iVal).*(e.^(-2*k2*a) )./(1+iVal)));
    BLABLA2 = ( 1 - ((1+iVal).*(e.^(2*k2*a) )./(1-iVal) ) ) - iVal.* (1 + ((1+iVal).*(e.^(2*k2*a) )./(1-iVal)));

    C = (e.^(-i*k1*a)).* ((e.^(-k2*a)./ ( BLABLA1 ) ) + (e.^(k2*a)./ ( BLABLA2 )));
    TChip = 4*(C.*conj(C))

    plot (E, [TSoln;TChip]);
     
    Last edited: Jan 23, 2016
  2. jcsd
  3. Jan 29, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Jan 29, 2016 #3
    Well, maybe I just need to work side my side with someone on this. I just can't seem to get to the answer..
     
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