Tunneling through step potential

[frankcastle]

Homework Statement

Hey, I found this interesting case in the tunneling problem.
How do we calculate the transmission probability when the
energy of the approaching particle is equal to the height of the potential barrier?

I.e E=Vo.

Homework Equations

Same equations as in other cases for x<0.

The Attempt at a Solution

How does the particle behave in the potential step region?

 

[najima]

can we calculate like the other problem of the potential barrier .E=V so after calculating Hamiltonian operator on wave function in part 1 we have Φ_1=Aexp(ikx)+Bexp(-ikx) and part 2 :Φ_2=0? and we can calculate A and B.

 

[frankcastle]

The solution for part 1 makes sense, but why must the wave function be zero in the square potential region? If tunneling is possible for E<V then transmission must occur for E=V too I think

 

[najima]

I got what was my mistake! :-)
General solutions are Φ_1=Aexp(ikx)+Bexp(-ikx) and Φ_2=Cexp(iqx)+Dexp(-iqx)
as we know k^2=2mE(4pi^2)/h^2 and q=2mE(E-V)(4pi^2)/h^2
so Φ_2=C+D (transmission is constant)it makes sense,doesn't it?;-)