Potential energy expectation value and virial theorem

I have been trying to solve this problem

Part a) can be proved using the formula .

For part b), the expectation value of the potential energy can be calculated from the integral ,

where

Therefore, the integral to be solved is

which gets simplified using spherical polar coordinates into

However, I have seen that the expectation value for the potential energy should be

Obtained using the same integral that I used, but using this expression for the wavefunction

which, from my understanding, should be the same expression but not in atomic units.

I am not familiar with conversions using atomic units, Bohr radius, Hartree, and so on. So I am not sure if what I have done is correct or if there's something missing.

For part c), using the virial theorem, the expectation value of the kinetic energy is

[T] = [E] - [V] = -0.5 - [V]

I am also confused by the Hamiltonian and the Laplacian which I did not use in my solution, am I missing something?

If I am not being clear please let me know and I will try to explain myself better. Thanks very much to whoever is willing to help me understand this problem better!

• I think the first step is to understand atomic units. In atomic units, the charge of the electron $e$ is set to 1, the Bohr radius is set to 1, and your energies will come out in Hartrees, without you having to do any conversions between eV and Hartrees. This will help with at least one thing: notice that when you write $e$ sometimes you're referring to the charge of the electron $\pu{1.602 \times 10^-19 C}$ and sometimes it's the base of the natural logarithm $2.718$. You can also use LaTeX-like syntax (MathJaX) to typeset equations here: $\frac{1}{2}$ $\to$ $\frac{1}{2}$ – orthocresol Jan 20 at 12:00
• I also presume that with part (a), you're not supposed to use that formula, but rather explicitly evaluate the integral $\langle \psi | \hat{H} | \psi \rangle$. – orthocresol Jan 20 at 12:03
• Thanks a lot for your reply! The elucidation about atomic units was very helpful. For the potential energy, e is the charge of the electron; for the wavefunction, e is Euler's number. I should stick to the exp notation to avoid confusion. Next time i'll use the LaTeX syntax, thanks. Yeah, for part (a) it makes more sense to evaluate the integral. The topic seems more clear now, I'll review the problem with these new notions. Thanks again for your help! – Jacob Jan 20 at 12:41
• if the potential $\sim r^n$ then $2T=nV$ where $T$ is average KE and V total potential energy and for coulomb potential $n=-1$ – porphyrin Jan 20 at 15:01