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·d^{2} =
ℏ^{2}÷2m. With d =
a_{0} = 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.
Instructions for use
The p-wave (l = 1) quantum defect for
sodium is D(1) = 0.86. Thus, the
quantum defect level structure predicts
E_{21} = –0.76947 Ry and
E_{31} = –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 g_{0} 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 E_{21} 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.
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
E_{0} 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.
Show the next-lowest-lying p-wave – designated
g_{1} 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 E_{31} 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.
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
|g_{0}(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.