Dirac delta spherical potential

[neworder1]

Three-dimensional particle is placed in a Dirac delta potential:

$$V = -aV_{0}\delta(r-a)$$

Find energy states and eigenfunctions for the angular quantum number $$l = 0$$.

 

[muppet]

There's a separate subforum for help with homework problems ("homework and coursework questions"). There's also a particular format in which you're expected to submit your questions in that subforum, including some attempt of your own to solve the problem.

 

[neworder1]

This is not homework :). Anyway, if someone moves the topic to that subforum, it's OK.

 

[malawi_glenn]

It is a HW related question, and you must do attempt to solution anyway.

 

[CPL.Luke]

I think he's doing it for fun out of summer boredom (its the wrong time of year for homework questions)

take a look at the 3 dimensional Schrödinger equation in spherical co-ordinates. its best to look at it in operator form.

also a nifty little trick for problems like this is to consider what your solutions ought to be. for a problem like this you clearly want your solution to be an eigenfunction of the L^2 operator (the traditional eigenvalues for this operator are chosen to be l(l+1) so you can qucikly conduct the separation of variables from there.

although I must say it would be much easier to solve this problem in terms of cartesian co-ordinates, although finding the l=0 states would be rather difficult.

 

[reilly]

Couple of things: 1. you have a Greens Function issue -- for example, go for E=0. Then you have the equation for a Coulomb potential from a charge of magnitude aVoW(a), where W is the wave function (Coulomb potential). More generally, you are dealing with a linear spatial wave equation, the Helmholtz Eq. with a source at a. Now, instead of a 1/r solution, you'll get ingoing and outgoing spherical waves. Again, the strength of the source is proportional to the wave function at a.

2. In momentum space, the problem appears to be a bit easier, or at least, more transparent. In fact, the momentum space version gives Cooper's Eq. for Cooper Pairs -- for an attractive interaction, a resonance for a + potential. You can see this from the consistency requirement that arises when W(a) occurs on both sides of the equation.

To find the l=0, or S wave component, simply integrate the wave function over the solid angle. Why?

There's an enormous amount of physics in this equation, including the core of the BCS theory of superconductivity, and a spectacular example of the HUP in action.

Regards,
Reilly Atkinson