- #1
Avatrin
- 245
- 6
Homework Statement
So, I start off with initial condition:
[tex]\psi(x,0) = e^{\frac{-(x-x_0)^2}{4a^2}}e^{ilx}[/tex]
This wavepacket is going to move from [itex]x_0 = -5pm [/itex] towards a potential barrier which is 1 MeV from x = 0 to x = 0.25 pm, and 0 everywhere else.
a = 1 pm
l = 2 pm-1
Homework Equations
I can use any numerical method to solve this. So, Euler's method, Runge-Kutta methods or Crank-Nicolson's method are all methods I can use.
Also, we have Schrodingers equation:
[tex](-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x))\Psi(x,t) = i\hbar\frac{d}{dt}\Psi(x,t)[/tex]
The Attempt at a Solution
I tried Euler's method:
[tex]\psi(x,t + \delta t) = \psi(x,t) + \frac{-i \delta t}{\hbar}\hat{H}\psi(x,t)[/tex]
Where:
[tex]\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)[/tex]
I used vectors (linspace). So, V, the second derivative of Psi and Psi were all vectors. However, Python kept casting away the imaginary parts off the elements in the vectors. So, my Psi's end up being constant for all t:
[tex]\psi(x,t + \delta t) = \psi(x,t)[/tex]
I don't know how to get past this.