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.

 

[DeeNos]

I understand your frustration with this problem and appreciate your willingness to seek help. Quantum chemistry is a complex field and it is common to encounter difficulties in solving problems.

Firstly, let me assure you that your trial function e^αr^2 is a valid approach for the 3-D, spherically symmetric, isotropic harmonic oscillator. The variational parameter α allows for the optimization of the wavefunction to better approximate the ground state energy of the system.

Looking at your Mathcad attachments, I can see that you have taken the correct approach in using the variational principle to find the ground state energy. However, there seems to be a mistake in your differentiation of the trial function. In the first attempt, you have taken the derivative of the entire function e^αr^2 instead of just the r^2 term. This leads to an incorrect expression for the derivative of the function. In the second attempt, you have taken the derivative of (d/dr function) instead of (d/dr function)*r^2, which also leads to an incorrect expression.

I would suggest going back to the basics and carefully differentiating the trial function e^αr^2 with respect to r to obtain the correct expression. I also recommend checking your algebra and ensuring that the final expression for the expectation value of the Hamiltonian is simplified correctly.

I hope this helps and wish you all the best in solving this problem. Remember, persistence and attention to detail are key in quantum chemistry. Keep up the good work!