Dirac Delta Function Potential (One Dimension)

[Mathemaniac]

Alright, I'm in my first QM course right now, and one of the topics we've looked at is solving the one-dimensional time-independent Schrödinger equation for various potentials, such as the harmonic oscillators, infinite and finite square wells, free particles, and last, but not least, the dirac delta function potential.

For each of these, with the exception of the dirac delta function, I can think of real-life situations that somewhat resemble these potentials. With the harmonic oscillator, we've got diatomic molecules. The potential wells are basically like containers. And free particles are obvious. (Well, these potentials are very rough ideas of what those real life situations would look like.)

But when it comes to the dirac delta function potential, I don't know of any real life situations that resemble it, and thus I cannot figure out why it is important to study it. The book we're using (Griffiths) doesn't seem to explain this. I asked my professor about this, but as I sit here during spring break while trying to think about his response, I guess I didn't really understand it or was satisfied with it. Come to think of it, it was so loaded that I don't even remember what the response was at this point!

So perhaps I could get some enlightenment on this issue here?

 

[CPL.Luke]

essentially the delta potential corresponds to an infinitely thin impenetrable wall classically.

it becomes important in qm as it allows you to look at things like transmission and reflection coefficients without a lot of the analytic clutter which comes with solving something like a finite step potential or a finite square well.

 

[Mathemaniac]

it becomes important in qm as it allows you to look at things like transmission and reflection coefficients without a lot of the analytic clutter which comes with solving something like a finite step potential or a finite square well.

Hmm, I guess that makes sense. I suppose that's why Griffiths begins the topic with a discussion on the difference between classical states (bounded, unbounded, scattering) and quantum ones.

On the other hand, with problem #2.34, he has us calculate the reflection coefficients for the finite step potential on our own. But such is the way of textbooks; they do the easy problems for examples and make you do the hard ones.

Anyway, thanks for the answer.

 

[CPL.Luke]

thas odd in my copy of griffiths he dos the finite step potential in the next session, calculating the reflection and transmission coefficients for such a potential is ather difficult and can take a very long time.

 

[Mathemaniac]

thas odd in my copy of griffiths he dos the finite step potential in the next session, calculating the reflection and transmission coefficients for such a potential is ather difficult and can take a very long time.

Really? He does the finite square well after right after the Dirac Delta potential, but then the step potential is left as a problem (#2.34, which is apparently one of those "must do" problems; I've yet to try it). This is the second edition I have. Chapter 2 is pretty lengthy as is, so he might have taken it out and left it as a problem.