Numerical Solutions to the Radial Schrodinger's Equation?

In summary, the author is researching numerical solutions to the radial form of Schrodinger's equation. They are trying to find a good method and are in need of guidance. They also want to present their work to a class.
  • #1
sciboinkhobbes
22
0
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 Schrodinger'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 Schrodinger'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!
 
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  • #2
sciboinkhobbes said:
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 Schrodinger'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 Schrodinger'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)
 
  • #3
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...
 
  • #4
sciboinkhobbes said:
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 Schrodinger 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.
 
  • #5
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 Schrodinger equation lead to the time independent version, then use conservation of angular momentum to get an effective potential and a one-dimensional Schrodinger equation with independent variable r. Then implement boundary conditions built into the L matrix to approximate the second derivative... At which point the Schrodinger 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:

[tex]- \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[/tex]

which, with the LJ potential, would be:

[tex]- \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[/tex]

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

1. What is the Radial Schrodinger's Equation?

The Radial Schrodinger's Equation is a mathematical formula that describes the motion of a particle in a central potential, such as an electron in an atom. It takes into account both the kinetic energy and the potential energy of the particle.

2. Why are numerical solutions used for the Radial Schrodinger's Equation?

Numerical solutions are used for the Radial Schrodinger's Equation because it is a complex equation that cannot be solved analytically. Therefore, numerical methods must be used to approximate the solution.

3. What are some common numerical methods used to solve the Radial Schrodinger's Equation?

Some common numerical methods used to solve the Radial Schrodinger's Equation include the Runge-Kutta method, the finite difference method, and the finite element method. These methods involve breaking down the equation into smaller parts and using iterative calculations to approximate the solution.

4. How accurate are numerical solutions to the Radial Schrodinger's Equation?

The accuracy of numerical solutions depends on the specific method used and the parameters chosen for the calculation. In general, the more iterations and smaller time steps used, the more accurate the solution will be. However, numerical solutions can never be exact and are always an approximation of the true solution.

5. How are numerical solutions to the Radial Schrodinger's Equation used in practical applications?

Numerical solutions to the Radial Schrodinger's Equation are used in a variety of practical applications, such as in quantum chemistry and materials science. They allow scientists to model and understand the behavior of particles in different environments and under different conditions, providing valuable insights into the properties and behavior of matter.

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