Schrodinger's Equation, Potential Energy Step E>U

[Oijl]

Homework Statement

Consider particles incident (in one dimension) on a potential energy step with E>U.
(That is, particles of total energy E are directed along in one dimension from a region of U=0 to a region of E>U>0.)

Apply the boundary conditions for \Psi and d\Psi/dx to find the probabilities for the wave to be reflected and to be transmitted.

Evaluate the rations [|B'|^2]/[|A'|^2] and [|C'|^2]/[|A'|^2] and interpret these terms.

Homework Equations

x<0: \Psi1 = A'e^(ikx) + B'e^(-ikx)
x>0: \Psi2 = C'e^(ik1x) + D'e^(-ik1x)

The Attempt at a Solution

I know that the boundary conditions are \Psi1(x=0)=\Psi2(x=0) and d\Psi1(x=0)/dx.

But how do I find the probabilities for the wave to be reflected or transmitted? Would that happen to be those ratios? Because if |B'|^2 = |A|^2, then the whole wave is reflected, so the probability of it being reflected is 1... AND the ratio would be 1.
So is that the right idea?

Thanks.

 

[Oddbio]

This is posted in the wrong section of the forum.

But, the idea is to model this as a realistic situation. If you were trying to measure the transmission and reflection you wouldn't be firing particles at both sides.
So basically you only have the A' coefficient, however because there could be transmission and reflection you will also need the coefficient for the reflected wave B' and the transmitted wave C'.
D' would be a particle fired AT the potential from the other side, but you don't consider that, so you set D' to zero.

So, now that you know A' is the initial amount that is fired at the potential, what do you think the ratios represent?

 

[Oijl]

Not the right section of the forum? Sorry; where should I have posted it?

The ratios are the ratios of reflected to initial, and transmitted to initial. This sounds like, then, the probability of the wave to be reflected and the probability of the wave to be transmitted. But if so, why am I asked to "find the probabilities for the wave to be reflected and to be transmitted" and then asked to evaluate and interpret the vary ratios I had just used to find the probabilities?

Or am I giving too much credit to the question-writer? English isn't his first language, anyway.

 

[Oijl]

Besides that when I solve for both A' and B' in terms of C and try to evaluate the ratio, I find
(k-k1)/(k+k1)
which is fine for talking about ratios, but when asked for a probability I want to give a numerical answer.

 

[Oddbio]

Actually I'm kinda new here myself, and considering that no moderator moved your thread then I guess this is the correct forum, my bad :)

I guess the author meant it as a hint then. However, I think it would be more straightforward from there if you found C and B in terms of A, instead of A and B in terms of C. Since you will be dividing by A.
Also, I didn't check if your answer is completely correct, but it looks right. It should be a function of k.
Try and think about why that is... what does k depend on?

and sorry for the delayed response.

 

[vela]

Not the right section of the forum? Sorry; where should I have posted it?

A good rule of thumb is that questions for lower-division classes should go here; questions from upper-division classes should go in the advanced physics forum.

Besides that when I solve for both A' and B' in terms of C and try to evaluate the ratio, I find
(k-k1)/(k+k1)
which is fine for talking about ratios, but when asked for a probability I want to give a numerical answer.

You can't give a numerical answer unless you have specific values for E and U. You can check various cases though. For example, if E>>U, you'd expect the particle to not even notice the small bump in potential and just go sailing through whereas if E<U, the particle should always be reflected. Are your results consistent with these scenarios?