The wave function for the hydrogen atom quantum state, which has and , is

in which a is the Bohr radius and the subscript on gives the values of the quantum numbers . (a) We have to show analytically that has a maximum at , and (b) we have to find the radial probability density for this state. (c) We have to show that

and thus that the expression above for the wave function has been properly normalized.

Solution:             Click For PDF Version

(a)

The wave function for the hydrogen atom quantum state, which has and , is

The square of the wave function will be

We will calculate next .

The condition

will determine the extremum of the function . We have

The solutions of the equation             Click For PDF Version are

, and .

One can show that is a minimum and is a maximum.

(b)

The radial probability density will be and its expression will be

(c)

We next check the normalization of the wave function by calculating the integral

We have

We evaluate the integral by making the following substitutions:

.

We get