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