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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)(d

^{2}[tex]\psi[/tex]/dx

^{2}) + 1/2kx

^{2}[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)(d

^{2}[tex]\psi[/tex]/dx

^{2}) + 1/2kx

^{2}[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 doesnt reach this level of detail.

Any help would be much appreciated!!