# Penetrating the Rectangular Potential Barrier, E < V0

1. Jan 23, 2016

### Chip

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. Jan 29, 2016