- #1

Saraphim

- 47

- 0

## Homework Statement

What is the probability that an electron in the ground state of hydrogen will be found inside the nucleus?

a) First calculate the exact answer, assuming the wave function [itex]\psi(r,\theta,\phi) = \frac{1}{\sqrt{\pi a^3}} e^{-r/a}[/itex] is correct all the way down to r=0. Let b be the radius of the nucleus

b) Expand your result as a power series in the small number [itex]\epsilon = 2b/a[/itex] and show that the lowest-order term is the cubic: [itex]P \approx (4/3)(b/a)^3[/itex]. This should be a suitable appoximation, provided that b << a, which it is.

## The Attempt at a Solution

Using partial integration a few times, I've gotten an answer for a) which I think is sufficient in order to solve b). [itex]P (r<b) = 1-e^{-2b/a}(2b^2/a^2 + 2b/a + 1)[/itex].

However, I'm completely lost on b). I'm looking for pointers here on how to even attack the problem. I can rewrite P so that I get [itex]P = 1 - e^{-\epsilon}(\epsilon/2 + \epsilon + 1)[/itex] but from there I'm stumped. I think the problem is that I've mostly forgotten how to do Taylor expansion, and a quick lookup does not help me in attacking this specific problem.