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- Copyright 1999, 2000 John S. Winn. All rights reserved. Last updated September 8, 2000.
- Ahead to hydrogen molecular ion orbitals.
The radial distribution function, rdf, (see Example 12.8 on page 431) measures the probability of finding the H atom's electron at a distance r from the nucleus, integrated over all angular coordinates. It is a function of the two quantum numbers n and l only, both of which you can change here (but don't go nuts - really huge values will either take forever to display, crash your browser, or produce graphs that are unreadable).
Related to the rdf is the notion of the effective potential for electron motion, as discussed on page 428 and shown in Figure 12.13. Classically, the electron moves along a line through the nucleus only for zero angular momentum states (l = 0). For l non-zero, the electron orbits (in an elliptical, not a circular, orbit) with well-defined distances of closest approach (minimum r) and maximum extent (maximum r). The graphs below show this classical region as a horizontal red line.
The left graph plotes the rdf for any chosen n and l values. The red bar spans the classically allowed range of motion, and the vertical gray line locates the expectation value for r. The right graph plots the rdf along with the effective potential V in blue. In this graph, the classical range of motion is also drawn at the total energy of the selected state (which depends on n only, of course), and the grayed areas denote classically forbidden tunneling regions. Count nodes in the rdf. These are spherical nodes, as defined on page 431.
Try this: select the value for n and change it to l + 1 so that you can enter various l values and automatically update n to one greater than l. As you increase l, note how the effective potential changes (and note that the graph axes automatically scale). You will see the rdf become more and more centered in the classical region, peaking closer and closer to <r>. The atom is behaving more classically as the quantum numbers increase. This large l limit approaches true classical orbiting at a fixed radius. In both graphs, r is measured in multiples of the Bohr radius. The rdf and V scales are the same as in Example 12.8 and Figure 12.13, respectively.
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