Exercise: The Thomas-Fermi Atom

The applet models stationary states for the valence electron(s) in heavy atoms. The choice of 'Flex Units' here means the scales for distance (d) and energy (E) are related as E·d² = ћ²/2m. With d = a₀ = 0.529 Å (1 bohr), E becomes the Rydberg energy, 1 Ry = 13.6 eV, when m is the electron mass. The valence electrons of multi-electron atoms move in the field of a screened Coulomb potential. For heavy atoms, screening effects are incorporated using the Thomas-Fermi model potential; in Flex Units this is V(r) = –2Zexp(–r/a)/r. To avoid the singularity at r = 0, the applet uses a truncated Thomas-Fermi potential – see 1st Technical Note below. V(r) and its associated parameters for the outer electron in a gold atom are recorded on the Equation View of the applet. On the tab labeled Graphics: [r] this potential is plotted over the interval [0, 5.0 bohrs]. The listing to the right of the graph includes a placeholder for the radial wave representing the 6s valence electron in gold.

Radial waves in the applet are indexed simply by the order in which they first appear, which differs from the conventional (nlm) labeling of hydrogen-like states.

1. Show the s-wave – designated g₀ in the applet – by right-clicking its placeholder in the list, clicking the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the "OK" button. The displayed waveform has no noticeable discontinuity, but in this case appearances are deceiving. Right-click anywhere in the graph and select "Display in Window" from the popup menu. This frees the graph to 'float' in full view while we make adjustments to the energy. Reposition the graph as desired and proceed to Equation View. Click in the value field for E₀, the energy for this s wave stationary state, and select "Edit parameter..." from the popup menu. Note the sizeable wave mismatch in the field labeled δψ — reliable results require this number to have magnitude much less than unity. Click the reset button [to re-position the match point – see 2nd Technical Note below] and observe the change in the display. Since quantum wavefunctions must be everywhere continuous, the energy of this state is not one of the allowed energies for the electron in gold.
2. Adjust the energy of this state upward (toward less negative values) to eliminate the discontinuity. Use the slider to change the highlighted digits in the text field while observing the waveform; the first digit highlighted can be moved left (right) using the up (down) arrows to the right of this field. [Extra digits can be added before or after the decimal by typing directly in the text field.] Reset the matching point once more when nearing the correct energy. When no discontinuity is evident, the energy is "allowed" and the wavefunction is one of the s-type radial waves for the electron in gold. Count the number of nodes to see which state you have found. The 6s radial wave has 5 nodes; if yours has fewer, continue searching at higher energies. Finish by selecting "Close", then "OK" to close all dialogs with the current settings.
3. Find the most probable distance of the 6s electron from the gold nucleus. The 6s radial wave displayed in the applet is actually the effective one-dimensional matter wave g(r) = rR(r) that directly furnishes the radial probability density as P(r) = |g(r)|². Thus, the most probable location of the 6s electron is that value of r that maximizes |g₀(r)|. Locate this distance by right-clicking anywhere on the graph, selecting "Trace" from the popup menu, and surveying the results. Since the 6s electron is the outermost electron in gold, it is reasonable to equate the most probable distance of the 6s state with the size of the gold atom. Compare this to the size of the hydrogen atom, using the same rule.

Technical Notes

• The truncated Thomas-Fermi potential used in the applet has the Thomas-Fermi form down to r = b and is constant therafter. The difference is inconsequential provided b is small. The default (b = 0.0001 bohr) is on the order of the nuclear size, below which the Thomas-Fermi form would not be expected to be accurate anyway.
• The Numerov method is used to construct stationary waves for a given (trial) energy. Using boundary conditions derived from the correct asymptotic form at r = 0, and then again at the interval endpoint, the Schrödinger equation is integrated inward to a preset match point. The right-side wave is then scaled to make its slope agree with that of the left-side wave at the match point. If the trial energy is allowed, the wavefunction values also will agree at the match point; otherwise, a discontinuity results. The [fractional] discontinuity is recorded in the tolerance text box labeled δψ. The technique is most accurate when the match point coincides with an extremum of the wavefunction. Each press of the reset button re-positions the match point to an extremum of the current waveform.