we can compute the radial wave functions Here is a list of the first several radial wave functions .

For a given principle quantum number
,the largest
radial wavefunction is given by

The radial wavefunctions should be **normalized** as below.

* Example:
Compute the expected values of , , , and
in the Hydrogen state
.*

The pictures below depict the probability distributions in space for the Hydrogen wavefunctions.

The graphs below show the radial wave functions. Again, for a given the maximum state has no radial excitation, and hence no nodes in the radial wavefunction. As gets smaller for a fixed , we see more radial excitation.

A **useful integral** for Hydrogen atom calculations is.

* Example:
What is the expectation value of in the state ?*

* Example:
What is the expectation value of in the state ?*

* Example:
What is the expectation value of the radial component of velocity in the state ?*

Jim Branson 2013-04-22