Transmission coefficient (quantum tunnelling)

[Camilla]

Homework Statement

What is transmission probability (T), when the incident particle energy (E) is equal to the energy of the barrier (U)?

Homework Equations

Equations to transmission and reflection probabilities.
http://i950.photobucket.com/albums/ad348/gs5720/img099.png?t=1278212132
http://i950.photobucket.com/albums/ad348/gs5720/img098.jpg?t=1278212130

The Attempt at a Solution

I know that the probability isn't null and less than 1, but i don't know how to proceed when the energy of E is equal to U.

That is all the information I have, I would appreciate any help.

 

[vela]

Try taking the limit of T as E approaches U₀. Use the fact that 1-U₀/E will approach 0 and approximate sin x by x when x ~ 0.

 

[graphene]

You'll need to solve the Schrödinger equation with E = Uo.

Don't try to find T by putting E = Uo in those two cases. This is because the solution to $$\frac{{d}^{2}\psi}{d{x}^{2}} = k$$ is different for the three cases where k is positive, negative or zero. You cannot solve the equation for positive k, and then put k=0 in the solution to obtain the solution for k=0.
So you'll have to solve Schrödinger equation again with E = Uo inside the barrier.