Hydrogen-like Wave Functions[1]

                Define

     (1.1)

The Hydrogen wave function separates into

             (1.2)

The last term is

          (1.3)

The middle term is

                (1.4)

The P in (1.4) is the associated Legendre function.   Press[2] gives a short stable code for calculating these.  He also gives a warning that "most of the recurrences involving m are unstable, and so dangerous for numerical work.  AssoLeg.htm .doc

                      (1.5)

The associated Laguerre polynomial is  

  There appear to be no warnings about this function.

The 1s wave function is

      (1.6) equation 1.6

The 2s wave function is

(1.7) equation 1.7

The 3s wave function is

(1.8)

 

Comment

                The ground state H wave function in the limit as r  infinity is

 

The 2s Li wave function in the limit as r  infinity is

 

The 3s Na wave function in the limit as r  infinity is

 

In each case the Z_eff  is on the order of 1, owing to the fact that the inner shells are filled.  Note that the higher order states are beginning to have a constant wave function in the asymptotic regions.

                In the K shell region Z_eff = 11.  In the L shell region Z_eff=9.  In the M shell region Z_eff = 1. 

Figure 1  Regions for various Z_eff's

The code for plotting the probabilities is for\plotsorbs.for. 

Figure 2  The K and L shell of sodium.  The M shell is a flat blue at the bottom of the plot.

Figure 3  Scale in y a factor of 12 larger and that in x a factor of 10 larger than in the above figure.

Assignment

 

The orbital energy of the 1s state is

(1.9)

This consists of three parts.  The probability distribution

 (1.10)

This is prepared for plotting on an even grid in for\plotsorbs.for.  The potential energy distribution

(1.11)

And finally. the Kinetic energy distribution is given by

(1.12)

Remember that the Z_eff is crude, define the derivative to be  

Approximate  to handle the negative value at the origin.  Use the Z_eff’s of figure 1.  E.g.

 

                Use the BLI to find P(r), V(r), and T(r).  Plot these and send me the plots.

A better approximation to Z_eff is described in Zeff.htm .doc