- #1

CAF123

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## Homework Statement

The normalized energy eigenfunction of the ground state of the hydrogen atom is ##u_{100}(\underline{r}) = C \exp (-r/a_o)##, ##a_o## the Bohr radius. For this state calculate

1)##C##

2)The radial distribution function, the probability that the electron is within a sphere of radius ##a_o##

3)Expectation values of ##r##, ##V(r)## and the uncertainty ##\Delta r##

## Homework Equations

Normalization of energy eigenfunctions, Expectation values.

## The Attempt at a Solution

The radial distribution function gives the probability of finding the electron in a spherical shell of thickness ##dr##. I understand that the RDF is commonly written ##4\pi r^2 R^2 dr##. However, is this the case here? When I do the integral I do not get the ##4\pi## (because of the state in question, the angular part squared gives a ##1/(4\pi)## which cancels)

My integral was $$\int_r^{r+dr} \int_0^{2\pi} \int_0^{\pi} u_{100}^* u_{100} r^2\, \sin \theta \,d\theta\, d\phi\, dr\, = \int_r^{r+dr} R_{10}^* R_{10} r^2\, dr,$$ which gives the probability of finding the electron between ##r## and ##r+dr## and the integrand is the RDF.

Many thanks.