The Discontinuity of Wave Functions in a Dirac Delta Potential

[hokhani]

consider a particle in one dimension. there is a dirac delta potential such as V=-a delat(x)
the wave functions in two sides(left and right) are Aexp(kx) and Aexp(-kx) respectively.
so the differential of the wave functions are not continious at x=0. what is the justification here?

 

[Vanadium 50]

Good question. What do you think the answer might be? What is the meaning of the derivative of the wavefunction?

 

[hokhani]

it is a criterion of the momentum and maybe they have opposite momentum. but i can't understand it exactly. i just know that one of the boundary conditions is the continuity of the differential of wave function!

 

[Vanadium 50]

If you want me to help, I will. But you're going to have to work with me. Write in proper English, and don't scream at me by putting a bunch of exclamation points at the end.

Go back a step - what is the physical meaning of the derivative of the wavefunction?

 

[hokhani]

ok. excuse me
i think the derivative of a wave function gives the momentum.

 

[Vanadium 50]

OK, good. Now, what's the derivative dp/dx of momentum?

 

[hokhani]

i think it is the kinetic energy of the particle.

 

[Vanadium 50]

No, it's not, but you're close. Let's come at it from a different direction - I think I'm confusing you. And please write in correct English. Capital letters, punctuation, the whole thing.

You say that the derivative of the wavefunction is momentum, and you are worried that the momentum is discontinuous. What would it mean if the momentum as a function of position were not continuous?

 

[hokhani]

sorry for my English and thank you for pointing that out.
my idea is that when the momentum direction is changed, we have a collision.

 

[hokhani]

Another problem:
Here the momentum operator P has complex eigenvalues while P is a Hermitian operator.

 

[Vanadium 50]

OK, so you have an abrupt change of momentum near a barrier. Is that a problem?

On to your second question - if you have a different question, you should ask it on a different thread. Since you refuse to use proper English, such as capital letters, you'll have to find someone else to help you.