• Quantum properties of the hydrogenic atom.

Pre-requisite skills:  An understanding of basic quantum physics and the meaning of the wave function and eigenvalue problem.

Approximate completion time:  Under an hour.

Provide sufficient detail to verify that the assignment was completed in a meaningful manner.

Applet by Wolfgang Christian

1.  This applet plots the probability amplitude, not the probability density, of the elecron of a hydrogenic atom as a function of the radial distance from the nucleus.  Notice that for l = 0 the probability amplitude is nonzero at the origin.  Because this system has spherical symmetry, the probability-amplitude-squared must be multipled by r, the radial coordinate, in order to obtain the probability density. 

(a)  What should be the probability density of the electron at the origin?  (In other words, what should be the probability of finding the electron in the center of the nucleus?) 

(b) Suppose we left off this r term and simply squared the radial wave function (the probability amplitude) plotted above in order to obtain the probability density.  What would be physically wrong with our result?

2. Set n = 5, l = 4, and m = 0.  Now increase m and examine the radial plot.  Explain why the radial wave function is said to be "m-independent."

3.   For a given value of n, the quantum number l can only take on certain values.  What are those values?   Use the applet to test your answer.

4.  For a given value of l, the quantum number m can only take on certain values.  What are those values?   Again, use the applet to test your answer.

5.  For the hydrogenic atom, the energy levels only depend on the principal quantum number n.  Since a wave function is defined according to all three quantum numbers, and not just n, this means that there are often a large number of different wave functions that correspond to the same energy level.   We call this number the degeneracy of the energy level.

(a)  For the energy level defined by n = 6, how many different wave functions correspond to this energy level?

(b)  Can you find a general expression that relates the degeneracy for a given quantum number n?

6.  A node is a point in the wave function that is 0, not counting the ends. 

(a) By changing the quantum numbers n and l and examining the resulting radial plot, what is the relationship between the number of nodes and the quantum numbers n and l? 

(b) Is the quantum number m involved in node-counting?   Explain.  (See Question 2 above.)

Helpful Resources

1. Vectors and Motion by Tom Henderson.
2. The Physics Hypertextbook by Glenn Elert (see Mechanics)
3. Physics E-Book - Projectile Motion by Fred Gram (formerly, Zogrseb, of the planet Ktoobirzp).
4. Book of Phyz - Motion by Dean Baird

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