The Discontinuity of Wave Functions in a Dirac Delta Potential

In summary, the conversation discusses a particle in one dimension with a Dirac delta potential and the wave functions on both sides of the potential barrier. The question is raised about the discontinuity of the derivatives of the wave functions at x=0 and the justification for this. The physical meaning of the derivative of the wave function is discussed, along with the potential for collisions and the problem of complex eigenvalues for the momentum operator. The conversation concludes with a suggestion to ask a different question on a different thread and a reminder to use proper English.
  • #1
hokhani
483
8
consider a particle in one dimention. 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?
 
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  • #2
Good question. What do you think the answer might be? What is the meaning of the derivative of the wavefunction?
 
  • #3
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!
 
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  • #4
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?
 
  • #5
ok. excuse me
i think the derivative of a wave function gives the momentum.
 
  • #6
OK, good. Now, what's the derivative dp/dx of momentum?
 
  • #7
i think it is the kinetic energy of the particle.
 
  • #8
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?
 
  • #9
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.
 
  • #10
Another problem:
Here the momentum operator P has complex eigenvalues while P is a Hermitian operator.
 
  • #11
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.
 
1)

What is a Dirac Delta Potential?

A Dirac Delta Potential is a mathematical concept used in quantum mechanics to represent an infinitely thin, infinitely tall potential barrier. This potential is often used to model interactions between particles or to represent a point-like particle.

2)

What is the significance of the Discontinuity of Wave Functions in a Dirac Delta Potential?

The Discontinuity of Wave Functions in a Dirac Delta Potential is significant because it represents a sudden change in the behavior of a particle when it interacts with the potential barrier. This discontinuity can affect the behavior and properties of the particle, such as its energy level and probability of being found in a certain location.

3)

How does the Dirac Delta Potential affect the wave function of a particle?

The Dirac Delta Potential causes a discontinuity in the wave function of a particle, meaning that the wave function is not continuous at the location of the potential barrier. This discontinuity can result in a change in the shape and behavior of the wave function, leading to different probabilities of finding the particle in different regions.

4)

Is the discontinuity of wave functions in a Dirac Delta Potential always present?

No, the discontinuity of wave functions in a Dirac Delta Potential is only present when the potential barrier is encountered by the particle. If the particle does not interact with the potential, the wave function will remain continuous and the discontinuity will not occur.

5)

What are some real-life applications of the Dirac Delta Potential?

The Dirac Delta Potential has applications in various fields, including quantum mechanics, solid-state physics, and nuclear physics. It can be used to model interactions between particles, such as in scattering experiments, and to study the behavior of particles in confined spaces, such as in quantum wells or nanoscale devices.

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