- #1
mjordan2nd
- 177
- 1
Homework Statement
This was a test question I just had, and I'm fairly certain I got it wrong. I'm confused as to what I did wrong, though. We were told that our potential was infinite when x<0, and Cx where x>0. We were asked to approximate the ground state potential using the variational method with a test function [tex]xe^{- \alpha x}[/tex].
Homework Equations
The variational method states that
[tex]\frac{\langle \psi \mid \hat{H} \mid \Psi \rangle}{\langle \psi \mid \Psi \rangle} \ge E_0.[/tex]
The Attempt at a Solution
I calculate
[tex] \langle \psi \mid \hat{H} \mid \Psi \rangle = \frac{3\hbar^2}{8m} + \frac{c}{4 \alpha^3}.[/tex]
and
[tex]\langle \psi \mid \Psi \rangle = \frac{1}{4 \alpha^3}.[/tex]
In total this gives me
[tex]\frac{\langle \psi \mid \hat{H} \mid \Psi \rangle}{\langle \psi \mid \Psi \rangle} = \frac{3 \hbar^2 \alpha^3}{2m} + c.[/tex]
Now assuming I had gotten this far correctly, I was a little confused where to go from here. Presumable I need to minimize this with respect to alpha. However, this would mean that alpha will be negative and unbounded, correct? Or do I need to assume that alpha is greater than 0. In that case, wouldn't the minimum energy just be c? This seems wrong to me. Any advice would be appreciated.