- #1
Kara386
- 208
- 2
Homework Statement
The Hamiltonian for an atom of deuteron is
##\hat{H} = \frac{-\hbar^2 \nabla_R^2}{2M} - \frac{\hbar^2 \nabla^2}{2\mu} - Ae^{\frac{-r}{a}}##
Where ##\nabla_R## is the differential operator for the centre of mass co-ordinates ##R = \frac{m_p\vec{r_p} + m_n\vec{r_n}}{M}## and ##\nabla## is the differential operator for the difference co-ordinates ##r_p## and ##r_n##. ##M## is the total mass ##m_p+m_n## and ##\mu## is the reduced mass ##\frac{m_pm_n}{m_p+m_n}##.
Approximate the ground state wavefunction with:
##\phi(\vec{r}) = ce^{\frac{-\alpha r}{2a}}## where c and ##\alpha## are real and positive constants.
1. Use the normalisation condition to find c in terms of a.
2. Find an expression for the expectation value ##<\phi|\hat H|\phi>##.
If there's a better way of doing Braket notation in latex, please let me know! :)
Homework Equations
The Attempt at a Solution
My first question is about 1. So the question has ##\phi## as a function of the vector ##\vec{r}##, but the r in the exponential is not a vector. So in what way is ##\phi## actually a function of ##\vec{r}##? I'm unclear on how that works, in terms of when I'm calculating the expectation value, am I integrating with respect to ##d^3\vec{r}##? Or just ##dr##, since there is no ##\vec{r}## dependence. If I do need to integrate w.r.t. ##d^3\vec{r}##, how do I do that? I have a feeling it involves spherical co-ordinates. Probably a ##4\pi r^2dr##. But I'd appreciate a good explanation as to why; I've had a look online and can't find one!
For the second question, the most horrible integral I've ever encountered appeared. The same question applies here, really: do I integrate ##d^3\vec{r}##? Or just ##dr##? Either way, I have no idea how to actually calculate ##\nabla_R^2## when both ##r## and ##\vec{r}## depend on ##R##. I really have no idea what ##\nabla_R^2 ## applied to ##\phi## actually means, in terms of how to calculate those derivatives!
That's a long question, most of it stating things I don't know, so thank you for taking the time to read it! And thanks for any help. :)