Exercise: Quantum Defect Model of the Atom

The applet models stationary states for the valence electron in monovalent atoms. The choice of Flex Units here means the scales for distance (d) and energy (E) can be adapted to the application at hand, but always are related as E·d² = ℏ²÷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 monovalent atoms, screening effects on the valence electron are incorporated using the quantum defect model; in Flex Units the potential energy is V(r) = –2(1 + b/r)÷r. To avoid the singularity at r = 0, the applet uses the truncated quantum defect potential (see 1st Technical Note below) that appears on the Math tab of the applet. On the tab labeled Graphics: [r] this potential is plotted over the interval [0, 20 bohrs]. The listing to the right of the graph includes placeholders for two p-type radial waves.

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

The quantum defect model asserts that this form for V(r) requires allowed energies given by (again in Flex Units) E_nl = –1/{n – D(l)}², where n is the usual shell label, l the orbital quantum number, and D(l) is the quantum defect. In this exercise we will confirm the quantum defect level structure for p-waves in the sodium atom.

1. The p-wave (l = 1) quantum defect for sodium is D(1) = 0.86. Thus, the quantum defect level structure predicts E₂₁ = –0.76947 Ry and E₃₁ = –0.21836 Ry for the energy of an electron in the 2p and 3p states of sodium, respectively. Show the first listed p-wave – designated g₀ in the applet – by right-clicking its placeholder, clicking on the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the OK button. In fact this state has the energy E₂₁ calculated above (switch to the Math tab to confirm), yet the waveform has a noticeable discontinuity, in violation of the rule that quantum wavefunctions must be everywhere continuous. The fault lies with the quantum defect screening parameter b, which is initialized to zero.
2. Adjust the screening parameter b upward to minimize the discontinuity. Go to the Math tab, where the screening parameter is recorded as b = 0. Right click anywhere in the value field for b and select "Edit Parameter..." from the popup menu to activate the Parameter Editor. Return to the Graphics: [r] tab and re-position the editor so as to afford an unobstructed view of the waveform. Use the slider to change the highlighted digits in the editor text field while observing the waveform. 'Fine tuning' is accomplished by adjusting the number of highlighted digits using the arrows to the immediate right of the text field.
   
   
   Extra digits can be added before or after the decimal by typing directly in the editor text field, then pressing the Enter key. The sum of leading and trailing digits is limited to 9.
   
   The correct screening value for sodium p-waves has been found when no discontinuity is evident in the radial waveform. For best results, reset the match point (see 2nd Technical Note below) as follows: right click in the value field for E₀ and select "Edit Parameter..." from the popup menu to open the Energy Editor, then click the reset button (but don't change the energy!). It may be necessary to reset the match point multiple times to achieve the smallest discontinuity. Now repeat the adjustment for b. Work back-and-forth between the two editors until you are satisfied that the discontinuity is minimal. Finish by selecting OK to end both edit sessions with the current settings.
3. Show the next-lowest-lying p-wave – designated g₁ 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 OK. Notice that this state has the energy E₃₁ calculated previously (return to the Math tab to confirm). With the value for b just found, this wave shows little or no discontinuity, thus verifying the quantum defect level structure for this case.
4. Find the most probable distance from the nucleus for an electron in the 2p state of sodium. The radial waves displayed in the applet are actually the effective one-dimensional matter waves g(r) = rR(r) that directly furnish radial probability densities as P(r) = |g(r)|². Thus, the most probable distance of the 2p electron from the sodium nucleus is that value of r that maximizes |g₀(r)|. Locate this distance by right-clicking anywhere on the graph, selecting "Trace On/Off" from the popup menu, and surveying the results.

Technical Notes

• The truncated quantum defect potential used in the applet has the quantum defect form down to r = a and is constant therafter. The difference is inconsequential provided a is small. The default (a = 0.0001 bohr) is on the order of the nuclear size, below which the quantum defect 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 editor tolerance field, 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.