What Is the Normalisation Constant for a Deuteron Wavefunction?

[ppyadof]

Homework Statement

This is the first question from a past exam paper I'm doing at the moment, and I'm not sure if it's a case that I'm doing something stupid, or if there is a problem with the question.

Q: The wavefunction of a deuteron can be approximated by:
$$\psi (r) = \frac{C}{r} e^{-\alpha r}$$
Where $\alpha = 0.23 fm^{-1}$Calculate the Normalisation Constant, C.

What is the probability that the separation of the two nucleons in the deuteron exceeds 2 fm, and what is their average separation.

The Attempt at a Solution

To work out C, I did:
$$\psi^{*}(r) = \frac{C}{r}e^{-\alpha r}$$
$$C^2 \int^{\infty}_{0} r^{-2} e^{-2 \alpha r} = 1$$

But if you work out this integral, it diverges, so what now?

As for the other part(s), am I right in thinking that you do something along the lines of:
$$P(r>2fm) = \int^{\infty}_{2*10^{-15}} \psi(r) \psi^{*}(r) dr$$
and
[tex]<r> = \int^{\infty}_{0} \psi(r) r \psi^{*}(r) dr [/itex][/tex]

 

[wisky40]

I think you shloud integrate over the whole space, so $$d^3x =r^2\sin\theta d\phi d\theta dr$$ because of the spherical geometry. However, in your case it should be $$r^2 dr$$ instead of $$dr$$ due to the radial part. I get $$C=\sqrt{2 \alpha}$$ and $$<r> = \frac{1}{2 \alpha}$$, but you should calcuate these results because I did the operations quickly.

I hope this helps.