Quantum rectangular barrier

1. Feb 9, 2007

judonight

Ok, so I've been dealing with this problem for a while and can't figure it out. (I tried to clean it up, but I don't know LaTeX, hopefully it is more clean in post #2; problem stated in #1, and my work in #2)

--Consider a (plane-wave) particle tunneling through a rectangular barrier potential w/ height v, and width a (the particle has E<v)....

a) write general solutions of stationary state functions in each 'region' of the potential.

--Done, no problem.

b)Find four relations among the five arbitrary constants in part a).

--Again, no problem

c) Use relations in part b) to equate transmission coefficient.

--Part c) is the problem, I have no idea what to do...

Last edited: Feb 9, 2007
2. Feb 9, 2007

judonight

1. The problem statement, all variables and given/known data

2. Relevant equations

a)

Region I: x<0 : psi(x)= A exp[i k x] + B exp [-i k x]

Region II: 0<x<a: psi(x)= C exp [K x]+ D exp [-K x]

Region III x>a : psi(x)= E exp [i k x]

$$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }i \hbar \pd{\Psi}{t}{} =- \frac{\hbar^2}{2 m} \ \pd{\Psi}{x}{2} + V \Psi$$

k^2= 2mE/hbar^2
K62= 2m(E-v)/hbar^2

c) took derivative of psi I = derivative psi II, at x=o.

Derivative psi II = derivative psi III, at x=a

psi(o)I=psi(0)II

psi(a)II=psi(a)III

3. The attempt at a solution

psi(0)I + d/dx psi(o)I = psi(o)II + d/dx psi(o)II

and same for when x=a for psi II and III.

I attempted to eliminate C and D constants, and solve for E. I let A =1 (thinking other waves than the incident have some multiple of amplitude.)

I thought that abs(E)^2= transmittion coeficient, is this not true? And B would be the reflection coefficient?

Where am I going wrong?

Last edited: Feb 9, 2007
3. Feb 9, 2007

The transmission coefficient represents the probability flux of the transmitted wave relative to that of the incident wave.

4. Feb 10, 2007

judonight

Ok, so the probability of incident wave over transmitted?

How do I get there from general solutions though?

(A*A)/(E*E) ?

5. Feb 10, 2007

judonight

I finally figured it out.

Eliminate coefficients of general solution for inside the barrier by applying boundry conditions at x=0, x=a.

Solve for coefficient of transmission, take abs, square it, bam, it's done.