Numerical Solutions to the Radial Schrodinger's Equation?

[sciboinkhobbes]

Hey everyone,

I'm starting a research project for my partial differential equations course, and I've chosen to research numerical solutions to the radial form of Schrödinger's equation. From some preliminary research, I've found information on using Numerov's method, but I am really not quite sure where to start.

Is Numerov's method the ideal way to approach this? Are there other good (or better) numerical methods? I'm assuming I'll be approximating the energy eigenvalues for the radial formulation of Schrödinger's equation (the given topic stated: "Solve the radial Schroedinger's equation for a central force other than Coulomb's law or Hooke's law. Perhaps the 6-12 rule?") I'm not sure what the 6-12 rule is either... I believe that I am supposed to write up a program in Matlab that will use some sort of algorithm (Numerov, Runge-Kutta?, etc...) to provide these solutions, but I am in desperate need of some guidance.

Having just started quantum mechanics myself, I'm not entirely sure how I should approach this problem, as I said, so any tips or information would be very much appreciated!

Thanks!

 

[jdwood983]

Hey everyone,

I'm starting a research project for my partial differential equations course, and I've chosen to research numerical solutions to the radial form of Schrödinger's equation. From some preliminary research, I've found information on using Numerov's method, but I am really not quite sure where to start.

Is Numerov's method the ideal way to approach this? Are there other good (or better) numerical methods? I'm assuming I'll be approximating the energy eigenvalues for the radial formulation of Schrödinger's equation (the given topic stated: "Solve the radial Schroedinger's equation for a central force other than Coulomb's law or Hooke's law. Perhaps the 6-12 rule?") I'm not sure what the 6-12 rule is either... I believe that I am supposed to write up a program in Matlab that will use some sort of algorithm (Numerov, Runge-Kutta?, etc...) to provide these solutions, but I am in desperate need of some guidance.

Having just started quantum mechanics myself, I'm not entirely sure how I should approach this problem, as I said, so any tips or information would be very much appreciated!

Thanks!

I'm pretty sure Numerov's method was designed for the radial solution to the SE, but if it wasn't designed for it, it certainly fits the criteria.

As for the 6-12 rule, maybe it is a reference to the Lennard-Jones potential. That could be it, but I'm not entirely sure because I've never heard of it expressed as a 6-12 rule (It has a potential that has a radius in powers 6 and 12).

If you give us some information as to what it is you are looking to do, I (or someone else) might be able to help direct you. What kind of presentation are you looking to do (is this just going to be a powerpoint presentation, a paper, or both?) And are you free to do it in any programming language? (some programming languages are easier than others)

 

[sciboinkhobbes]

Thanks so much for your response!

For the class presentation, my professor stated he was looking for either a 5 page paper (using the blackboard to write equations as needed), or a Powerpoint presentation with an attached appendix detailing code and procedure. I'm not sure, but for this project it seems that it would be best to write a paper, and then throw together a powerpoint for the class presentation... But I'm not sure if both are acceptable, haha. If it isn't, I'm leaning toward the paper.

There isn't a specific recommendation for programming language, but the only program I really have any familiarity with is Matlab, so I figured that's the way I would attack the problem...

 

[jdwood983]

Thanks so much for your response!

For the class presentation, my professor stated he was looking for either a 5 page paper (using the blackboard to write equations as needed), or a Powerpoint presentation with an attached appendix detailing code and procedure. I'm not sure, but for this project it seems that it would be best to write a paper, and then throw together a powerpoint for the class presentation... But I'm not sure if both are acceptable, haha. If it isn't, I'm leaning toward the paper.

There isn't a specific recommendation for programming language, but the only program I really have any familiarity with is Matlab, so I figured that's the way I would attack the problem...

Okay...so you probably want to make some sort of outline as to what it is you want to describe. You might want to keep in mind some of the following questions
--Are you going to use the Lennard-Jones potential or use some other (central) potential?
--What is it you want to show in your plots/solution (also what do you plan on plotting--MATLAB is excellent on its calculation & plotting abilities)
--How accurate are your results (possibly finding experiment results for the potential you are choosing)

I haven't tried using MATLAB for a numerical solution to the Schrödinger equation, but is there a package to solve the differential equation using Numerov's method? I know the Runge-Kutta 4 method is on there (ode45) but I haven't come across Numerov's method. If it does not exist, there may be a lot of work (possibly publishable?) involved in writing a MATLAB code that runs the Numerov method. You may want to consult with your professor on that particular differential equation method.

 

[sciboinkhobbes]

After speaking with my professor, he clarified that the "6-12" rule is indeed the Lennard-Jones Potential, and recommended that I use that as my central potential.

As opposed to Numerov's method, his suggestion was to describe how separation of variables in the time-dependent Schrödinger equation lead to the time independent version, then use conservation of angular momentum to get an effective potential and a one-dimensional Schrödinger equation with independent variable r. Then implement boundary conditions built into the L matrix to approximate the second derivative... At which point the Schrödinger equation becomes a generalized eigenvalue problem.

However, we were given the L matrix as a Mathematica notebook (tridiagonal matrix approximating the Laplacian with zero Dirichlet boundary conditions), but I'm not entirely sure how to use it.

At the moment, however, I'm taking a look at the radial equation:

- \frac{\hbar^{2}}{2m} \frac{d^{2} u}{dr^{2}} + \left[ V + \frac{\hbar ^{2}}{2m} \frac{l(l+1)}{r^{2}}\right] u = E u

which, with the LJ potential, would be:

- \frac{\hbar^{2}}{2m} \frac{d^{2} u}{dr^{2}} + \left[ 4\epsilon \left[(\frac{\sigma}{r})^{12} - (\frac{\sigma}{r})^{6} \right] + \frac{\hbar ^{2}}{2m} \frac{l(l+1)}{r^{2}}\right] u = E u

I think? Any suggestions from this point forward would be incredibly appreciated.