CHM 3411, Dr. Chatfield, Spring 2009

Problem Set 6

Due Friday, Feb. 27

 

Suggested (b) Exercises from Ch. 10: 1,3,2,4.  Discussion questions: 1,2,3,4 6.

 

This problem set explores the wavefunctions and the energies for hydrogenic atoms.  Problems 1-4 focus on calculations based on the wavefunctions (orbitals), and problems 5-6 focus on spectroscopy.

 

1. Determine rmp, the most probably distance of the electron from the nucleus, for an electron in a 2s orbital of a hydrogen atom.  [Hint: there is only one answer.  If you think there should be more, draw a picture of the radial distribution function for the 2s orbital, and you will see why there is only one.]

 

2. Calculate the expectation value <r> for an electron in a 2s orbital and in a 2pz orbital of hydrogen.  Which is larger?  Is this what you expected?

 

3. I RECOMMEND USING ATOM UNITS FOR THIS PROBLEM.  Calculate <V>, the expectation value for the potential energy, for an electron in a 2s orbital of a hydrogen atom.  Once you have calculated, <V>, calculating <T> (T is kinetic energy) is trivial since <T> = <E> - <V>.  How are <V> and <T> related?  How does this relate to the virial theorem (see p. 296)?

 

4. You should be able to answer the following question (in atomic units) in your head or with trivial calculations.  Consider the following unnormalized eigenfunction for the hydrogen atom:

 

 

a.       How many radial nodes does this function possess?

b.      What are the l and ml quantum numbers for this state?

c.       What is the energy of this state?

d.      What is the classical turning radius for this state?  [Recall that for the harmonic oscillator, we defined the classical turning point as the distance at which E=V.  A classical particle cannot enter the region beyond this point.  The classical turning radius is defined analogously.  It is the distance for which E=V; if the electron were classical, it would not move beyond this distance.]

 

5. Atkins Problem 10.2.

 

6. Atkins Problem 10.25.  The point here is that spectroscopy is a powerful tool for learning about the composition of starts.  You will see that two different isotopes of He+ emit light of slightly different wavenumber.  You are to calculate the wavenumbers for the specified electronic transitions for two different isotopes of He+.