# Removing units from Schrodingers equation

1. Apr 11, 2015

### Avatrin

1. The problem statement, all variables and given/known data
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 Schrodinger 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.

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

$$-V<E<0$$

3. 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 Schrodinger's equation and $\psi(x,t)$ unitless. So, I would love all advice that can potentially lead me to the solution.

Last edited: Apr 11, 2015
2. Apr 11, 2015

### robphy

Last edited by a moderator: May 7, 2017
3. Apr 11, 2015

### 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.

Last edited: Apr 11, 2015
4. Apr 11, 2015