Quantum Chemistry: approximations using trial functions

[RubberBandit]

Homework Statement

For a 3-D, spherically symmetric, isotropic harmonic oscillator, use a trial function e^αr^2 as a variational parameter.

The Hamiltonian as given by the book is attached, as are two files showing screenshots of my attempt at the solution in Mathcad.

I've tried it two ways. In addition to the method I've attached, in which I take the derivative with respect to r of (r^2*(d/dr function)), I also tried (d/dr function)*(r^2)(d/dr function). Neither have worked.

I'm just looking for another set of eyes to look over the math and see where I made an error. I have the solution (it's 3/2 (h/2∏)(k/μ)^1/2. I want to get to the solution myself, I've just been banging my head against this problem for hours now.

The mathcad attatchments are screenshots, so I hope they don't violate the "No scanned photos" policy.

Thanks for any help.

 

[DrClaude]

In the first equation ("What I'm solving"), you forgot the wave function after the potential energy (just inside the last closing bracket):
$$
\frac{k}{2} r^2 e^{-\alpha r^2}
$$

 

[RubberBandit]

Thank you!

Dr. Claude, thank you! That worked! I don't know how I kept missing that, but I did. Thanks for taking a look.

 

[DrClaude]

Glad to be of help.