Solving Coulomb Potential with Schrodinger Equation - Roger Newton & Mathematica

[FunkyDwarf]

Hey guys,

Am reading up on solving the Coulomb potential in the Schrodinger equation and have a few questions. Essentially what i want to do is check that by just plodding along using DSolve in Mathematica i can get the same result that people ages ago did by using fancy analytic tricks, ie make sure my computational method works so i can apply it elsewhere.

Anyway: in the book "Scattering theory of waves and particles, 2nd edition" by Roger Newton he plugs in a coulomb potential and gets a Whittaker function for the radial wavefunction (using partial wave method) which is fine because when you DSolve the Schrodinger equation with a coulomb potential of strength V0 you get two Whittaker functions, one of which i threw away because not only was its probability integrand unbounded at the origin but the derivative of the wavefunction was not zero at the origin (working in spherical polars here, radial wave function etc).

So, in my method, I'm left with an un-normalized Whittaker function which is, apart from the normalization, the same Whittaker function Newton gets so all seems well so far. Now as far as I am concerned, I'm done, because if i want a phase shift, which is ultimately what I am after, i can (to get a rough idea) numerically compare the solution at large r with the shifted free particle solution.

However, in the book they go further which i would understand if it was simply an attempt to get a closed form solution for the phase shift (which ultimately i don't care about as the problem i wish to solve can only be done numerically, but i'd like to make sure i can get the usual solutions) but after expressing the Whittaker function in terms of Laguerre functions he says:

"Similarly, we may define an irregular solution f_l (k,r), except that the boundary condition (12.15) no longer serves." Where 12.15 states that $$lim_{r->\infty}e^{ \pm ikr}f_{ \pm}(k,r)=1$$ which i guess i understand. He then says "It is easy to see from the Schrodinger equation that the closest we can come to it is to demand that:
$$lim_{r->\infty}e^{-i(kr-n ln(2kr))}f^{(c)}_{l}(k,r)=1$$ which i don't really understand.

Anyway, i guess the point of my question is: why are we entertaining the thought of including the irregular solution again? Didn't we discount it for a reason? Ultimately the problem for me is if i just plot the Whittaker function i get vs their asymptotic solution the phases are remarkably different, which is of course, a problem :)

If it would help I can scan the pages from the book that deal with this.

Hope that all made sense! (and that someone can point out my undoubtedly stupid error)
Cheers
-G

 

[eljose79]

regDear Greg,

Thank you for your forum post and for sharing your questions about solving the Coulomb potential in the Schrodinger equation. It is great to see that you are using computational methods to check the results obtained by analytical techniques, and I am happy to help you understand the concept of irregular solutions in this context.

Firstly, let me address your question about why we are entertaining the thought of including the irregular solution again, even though we discounted it initially. The reason for this is that the irregular solution, although not physically meaningful, still plays a role in the solution to the Schrodinger equation. This is because the irregular solution is a mathematical artifact that arises when solving the equation using the partial wave method, and it is related to the divergence of the Coulomb potential at the origin. Therefore, even though it is not a valid physical solution, it still affects the overall solution.

Now, regarding the expression for the irregular solution in terms of Laguerre functions, this is a mathematical trick that is used to simplify the solution and make it more manageable. By expressing the Whittaker function in terms of Laguerre functions, we can obtain a closed-form solution for the phase shift, which is ultimately what we are after. This is why the author goes further and includes the irregular solution in the solution.

As for your question about the boundary condition, it is important to note that the boundary condition you mentioned (12.15) is only valid for the regular solution, not the irregular one. Therefore, we need to modify it for the irregular solution, which is what the author has done by introducing the term involving the natural logarithm.

In conclusion, including the irregular solution in the solution is necessary for obtaining a closed-form solution for the phase shift, and it is also a way to account for its influence on the overall solution. I hope this helps clarify your doubts, and please do not hesitate to ask for further clarification if needed. Keep up the good work with your research!