- #1
razidan
- 75
- 1
Homework Statement
I am trying to figure out the average penetration depth into a finite step potential, similar to this thread:
https://www.physicsforums.com/threads/qm-barrier-penetration.104641/
But instead of just estimating the depth as ##\frac{1}{\kappa}##, i would like to calculate it explicitly.
Homework Equations
##<x>=\int \psi^* x \psi##
The Attempt at a Solution
I took the solution to the step potential problem - ##\psi=t \cdot e^{\alpha x} \text{ where } \alpha=\sqrt{2m(V-E)}/\hbar^2## .
when I evaluated the integral i have: ##<x>=\frac{k^2}{\alpha^2} \frac{1}{k^2 + \alpha^2}## where ##k=\sqrt{2m(E)/\hbar^2}## .
this has unit of meter squared, and this is because the wavefunction is not normalized, which leads me to my question: Can i normalize my wavefunction only on [0,##+\infty##]? (on the negative half of x, it cannot be normalized.
If i do so, I get ##|c|^2=2\alpha##.