Step Potential and Wave Functions

[erok81]

Homework Statement

I have a step potential where E < V₀. I also have to do this for E >V₀ but I think after we do the first part it'll make sense.

Plot the behavior of the wave function in crossing the region at x = 0, considering
four values of the energy. So I'll end up with four plots.

Homework Equations

n/a

The Attempt at a Solution

My problem is I don't know of four wave functions at x=0. I may be reading the problem wrong. Here is what I have so far. Which seems way too simple, so I am probably doing it wrong.

For the incident wave I have.

[tex] \psi (x) = Ae^{ikx}[/tex]

For the reflected wave.

[tex] \psi (x) = Be^{-ikx}[/tex]

And the transmitted wave.

[tex] \psi (x) = Ce^{- \alpha x}[/tex]

Am I missing some wave functions? The plotting I can do, but I don't think what I have is correct.

 

[anathema]

Hello,

Thank you for sharing your attempt at a solution. Your approach is on the right track, but there are a few things you may want to consider.

First, for the incident and reflected waves, you are correct in using the general form \psi (x) = Ae^{ikx} and \psi (x) = Be^{-ikx}. However, for the transmitted wave, the form you have used is for a decaying wave, which may not be appropriate for this problem. Instead, you may want to consider using the form \psi (x) = De^{ikx}.

Second, for the four values of energy, you can choose any four values that satisfy the condition E < V0. For example, you could choose E = V0/2, E = V0/3, E = V0/4, and E = V0/5. This will give you four different plots to compare.

Lastly, when plotting the wave functions, it may be helpful to also plot the potential barrier V(x) on the same graph. This will give you a better understanding of how the wave functions behave in relation to the potential barrier.

I hope this helps and good luck with your solution!