# Position vs. Harmonic basis for solving S.E.

1. Apr 10, 2007

### 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?

2. Apr 10, 2007

### 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.

3. Apr 10, 2007

### 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.

4. Apr 11, 2007

### 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.

5. Apr 11, 2007

### 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?

6. Apr 11, 2007

### christianjb

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.

7. Apr 12, 2007

### Meir Achuz

A variational calculation might be easier.