Scaling when solving Schrodinger equation numerically

[dilloncyh]

I guess this is just a maths problem about algebra. I'm learning to solve Schrödinger equation numerically, and right now I'm just dealing with the simplest examples like harmonic potential, square well, etc. The problem is that sometimes my program gives some strange results and I suspect it is deal to the extremely small values of the constants involved. So let's say I set hbar^2/m = 1, express the length scale in terms of Bohr's radius (5.3e-11 becomes 1) and express all the energy (E and V) in terms of eV. I need to first find the eigenvalue E that satisfies the equation, then I can use the value of E to find the initial conditions which enable me to find solution everywhere. But how do I convert E and all the values of wave function at different points back to some sensible units? I just do not know what the units of the results I get mean.

thanks

 

[Simon Bridge]

How did you "express all energy in terms of eV"?
(It looks a bit like you have over-specified your units.)

Note: If $\hbar^2/m = 1$ then you particle at rest has energy: $mc^2 = \hbar^2c^2 = 197.33\text{keV}$.

Units are commonly defined by the distance, mass, and time scales.

If you set c=1 then the mass of your particle is the unit of energy. If you put m=1 then your energies will all be in "particle mass-energy units".
If your particle is an electron, say, then multiplying by 511 will get the energy converted to keV.

Doing this has consequences:
If you also set unit-distance to, say, the bohr radius, then the unit of time becomes the duration for light to travel the bohr radius.
That's pretty small - but quantum stuff can happen on small time scales and you are more interested in stationary states anyway.

 

[Dr. Courtney]

In atomic physics, we solve this problem with atomic units, defining m, e, hbar, and k all to be 1.

https://en.wikipedia.org/wiki/Atomic_units