Position vs. Harmonic basis for solving S.E.

[christianjb]

I've monkeyed up a code which solves the 2D S.E. in a harmonic basis- i.e. writing the wf as a linear combination of harmonic oscillator states.

Has anyone got any references/comments on the relative accuracy/efficiency/drawbacks of using a harmonic basis instead of using the more direct position basis?

 

[Meir Achuz]

I don't know of a reference, but my guess would be that the harmonic basis is better for a confining potential, as with quarks.
If the large distance behavior of the wave function, were exp{-x/a},
then I think the harmonic basis would not be so good.
Most quark calculations use a harmonic basis.

 

[MaverickMenzies]

In my experience, (in quantum optics) using the fock (i.e. eigenstates of the simple harmonic oscillator) basis to solve the SE can be messy depending on the Hamiltonian. For example, a beam splitter transforms fock states in a complicated manner, but its action on quadrature eigenstates (position and momentum eigenstates) is more transparent.

 

[christianjb]

Thanks for the replies.

I guess if it's good enough for doing quarks then it ought to be good enough for me!

I'd still like to see a reference looking at the numerical aspects- error analysis etc.

My system is near harmonic- so it makes sense to me to use harmonic basis functions. I also like the fact that the kinetic energy matrix elements can be calculated analytically in this basis.

One disadvantage I have encountered is that the position grid spacing isn't always large enough to accommodate larger harmonic eigenfunctions, so I get some truncation error.

 

[Edgardo]

Hello christianjb,

would you mind giving a small tutorial on how you solved the Schrödinger equation?

For example what programs did you use? Maybe you could post the sourcecode?

 

[christianjb]

Hello christianjb,

would you mind giving a small tutorial on how you solved the Schrödinger equation?

For example what programs did you use? Maybe you could post the sourcecode?

Solving the 1D or 2D S.E. is not a particularly difficult problem once you get familiar enough with manipulating QM expressions.

Perhaps the most straightforward way is to use the familiar position basis (x-basis). That makes it easy to put the potential elements into the Hamiltonian, but you've got to use a finite difference scheme to put the KE elements in (i.e. evaluating the d^2/dx^2 operator).

I like using a H.O. basis because the KE matrix elements can be worked out analytically. However, the trade-off is that you then have to calculate the integrals over the basis functions of the PE terms.

Whatever your approach- you'll need to solve an eigenvalue/eigenvector eqn. at some point. I use the 'jacobi' subroutine from Numerical Recipes to do that.

Not a very good answer. I'm a bit pressed for time right now- but try starting up a new thread if you want to know the details and various peoples' approaches.

I'm sure there are lots of pages on the internet that give step by step instructions.

 

[Meir Achuz]

A variational calculation might be easier.