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Removing units from Schrodingers equation

  1. Apr 11, 2015 #1
    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 [itex]\psi(x,0)[/itex] which is the symmetric solution to the time-independent Schrodinger equation for the well. Then I can use Euler's method to find [itex]\psi(x,t)[/itex] for the other values of t (for an animation), ie, I use:

    [tex] \psi(x,t+dt) = \psi(x,t) + dt*\hat{H}\psi(x,t)[/tex]

    Where [itex] \hat{H}[/itex] 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:

    [tex] \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} [/tex]
    [tex]A=\frac{e^{\kappa a}\cos{la}}{\sqrt{a+1/\kappa}} \quad{\rm }\quad B=\frac{1}{\sqrt{a+1/\kappa}}[/tex]

    Here a is half the width of the potential square well, and [itex]\hbar[/itex] 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
    [tex] -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = i \hbar\frac{d\psi}{dt} [/tex]

    [tex]-V<E<0[/tex]

    3. The attempt at a solution
    I can see the relation between the third of and fourth constants; [itex]\sqrt{\frac{-V}{E} - 1}[/itex]. 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 [itex]\psi(x,t)[/itex] unitless. So, I would love all advice that can potentially lead me to the solution.
     
    Last edited: Apr 11, 2015
  2. jcsd
  3. Apr 11, 2015 #2

    robphy

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    Last edited by a moderator: May 7, 2017
  4. Apr 11, 2015 #3
    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 ([itex]|\psi|^2[/itex]). Does it look right?
    It changes into the same curve no matter what [itex]\psi(x,0)[/itex] is.

    movie.gif
     
    Last edited: Apr 11, 2015
  5. Apr 11, 2015 #4
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