Removing units from Schrodingers equation

[Avatrin]

Homework Statement

I am working on a problem regarding an electron in a one dimensional finite square well. I start off with $\psi(x,0)$ which is the symmetric solution to the time-independent Schrödinger equation for the well. Then I can use Euler's method to find $\psi(x,t)$ for the other values of t (for an animation), ie, I use:

$$\psi(x,t+dt) = \psi(x,t) + dt*\hat{H}\psi(x,t)$$

Where $\hat{H}$ is the hamiltonian.

However, I am encountering a problem:
I think the values outside of the well are too small for Python; When I use the actual values for the speed of light, the electron mass and Planck's constant, I do not get an animation that makes sense; Actually, I do not get any animation at all. However, when I set all of the mentioned values equal to one, while the animaton still does not seem correct, it at least gives me an animation.

So, I am thinking of making my equations unitless. However, I do not know how. I have the following six constants:

$$\frac{V}{\hbar}\quad{\rm}\quad \frac{\hbar}{2m} \quad{\rm }\quad \kappa := \frac{\sqrt{-2mE}}{\hbar} \quad{\rm }\quad l:= \frac{\sqrt{2m(E+V)}}{\hbar}$$
$$A=\frac{e^{\kappa a}\cos{la}}{\sqrt{a+1/\kappa}} \quad{\rm }\quad B=\frac{1}{\sqrt{a+1/\kappa}}$$

Here a is half the width of the potential square well, and $\hbar$ is Planck's reduced constant. E is the particles energy and V is the depth of the potential well. Finally, m is the mass of an electron.

Homework Equations

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = i \hbar\frac{d\psi}{dt}$$

$$-V<E<0$$

The Attempt at a Solution

I can see the relation between the third of and fourth constants; $\sqrt{\frac{-V}{E} - 1}$. The other constant, however, I am not certain how to make unitless.

Also, of course, there is no guarantee that my program will work when I make Schrödinger's equation and $\psi(x,t)$ unitless. So, I would love all advice that can potentially lead me to the solution.

 

[robphy]

Python or any other language should be able to handle those constants.
It is more likely that your step size is not commensurate with characteristic values of length and time in your problem.

However, there is value to making the problem dimensionless.
Possibly useful:
http://en.wikipedia.org/wiki/Nondimensionalization
http://www.kw.igs.net/~jackord/bp/h5.html
http://www.physics.csbsju.edu/QM/square.03.html

 

[Avatrin]

You may be right. However, what should the step size be? In picoseconds?

Also, here is my animation when I changed the units to one ($|\psi|^2$). Does it look right?
It changes into the same curve no matter what $\psi(x,0)$ is.

 

[Sergey Vilov]

Usually such simulations are performed in reduced(atomic) units:http://en.wikipedia.org/wiki/Atomic_units