- #1
elninio0397
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Homework Statement
A particle with energy greater than the potential is defined as below:
V(x) = Vo (x<0)
V(x) = 0 (0<x<a)
V(x) = Vo (x>a)
a) Write the complete solutions (time and space) to the S. Eqn for the 3 regions
b) What condition must the width of the potential satisfy for the transmission of a wave from the left to be a maximum?
c) What is the minimum possible value for the transmission? What conditions must the weidth of the potential satisfy for this?
Homework Equations
None, really. It seems more conceptual.
The Attempt at a Solution
If I am not mistaken, this represents a simple harmonic oscillator.
I believe to have figured out the answer to part A.
Region 2 has the Schrodinger equation:
(-h(bar)/2m)(d2[tex]\psi[/tex]/dx2) + 1/2kx2[tex]\psi[/tex] = ih(bar)d[tex]\psi[/tex]/dt
whose bound state solutions are [tex]\psi[/tex](x,t)=[tex]\psi[/tex](x)e^(-iEt/h(bar))
The solution for Region 1 = Region 3 is:
(-h(bar)/2m)(d2[tex]\psi[/tex]/dx2) + 1/2kx2[tex]\psi[/tex] = ih(bar)d[tex]\psi[/tex]/dt
[tex]\psi[/tex]''(x)=(alpha)2[tex]\psi[/tex](x) where (alpha)2 = (2m/h(bar)2)(V(x)-E)
For problem b, I have an idea that since the wave function is Asinkx, the length (x) would have to be long enough so that it reaches a maximum at the second barrier (sin(pi/2)).
I could be going about this problem completely wrong. I've read through the textbook many times and it just doesn't reach this level of detail.
Any help would be much appreciated!