HQ3 Hydrogen atom: mean and mode

Write down the definition of the expectation value $\langle g(x) \rangle$ of some function g of a random variable X. Explain what it means.

Write down the definitions of the mean and the standard deviation of X. Explain what they mean.

The PDF for the radial separation r of the electron from the proton in the lowest-energy state of a hydrogen atom is prescribed by quantum mechanics as:

$$f(r) = \frac{4}{a_0^3}r^2 e^{-2r/a_0}$$

where a₀ is a length (the Bohr radius, $5.29 \times 10^{-11}\, m$).

1.

Sketch the PDF.

2.

Show that it satisfies the normalisation condition ( KP2.9 ).

3.

Show that

$$\langle r \rangle = \frac{3a_0}{2}$$

4.

Find the most probable value of r.