Using the normalisation condition in 3D

[Kara386]

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. :)

 

[blue_leaf77]

If there's a better way of doing Braket notation in latex, please let me know! :)

Use \rangle, $\rangle$ or \langle, $\langle$.

My first question is about 1. So the question has ϕϕ\phi as a function of the vector ⃗rr→\vec{r}, but the r in the exponential is not a vector. So in what way is ϕϕ\phi actually a function of ⃗rr→\vec{r}?

A notation of the form $\phi(\vec{r})$ can also be written as $\phi(r, \theta,\varphi)$, therefore if only $r$ is present in the argument, that means $\phi$ is independent of $\theta$ and $\varphi$.

I'm unclear on how that works, in terms of when I'm calculating the expectation value, am I integrating with respect to d3⃗rd3r→d^3\vec{r}? Or just drdrdr, since there is no ⃗rr→\vec{r} dependence. If I do need to integrate w.r.t. d3⃗rd3r→d^3\vec{r}, how do I do that?

You are integrating with respect to $d\vec{r}^3$. It's just the notation for the volume element in spherical coordinate. So, find out the expression for the volume element in this coordinate system.

But I'd appreciate a good explanation as to why; I've had a look online and can't find one!

You can as well integrate in Cartesian coordinate, but since the given form of the wavefunction is easier to be integrated in the spherical coordinate, you better do it this way.

Either way, I have no idea how to actually calculate ∇2R∇R2\nabla_R^2 when both rrr and ⃗rr→\vec{r} depend on RRR. I really have no idea what ∇2R∇R2\nabla_R^2 applied to ϕϕ\phi actually means, in terms of how to calculate those derivatives!

No, $\vec{r}$ and $\vec{R}$ are independent variables. The question is rather strange actually because it asks you to calculate the expectation value of the total energy but it gives you only the wavefunction of the relative position. In this case, I would suggest that you define your own wavefunction for the center of mass. Note that the total wavefunction should look like $\psi(\vec{R},\vec{r}) = \gamma(\vec{R})\phi(\vec{r})$ where $\gamma(\vec{R})$ is the wavefunction of the center of mass. To find this wavefunction, note that the total Hamiltonian suggest that it should be that of a plane wave.