The applet above 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) are related as E·d^{2}
= ћ^{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 Technical
Note #1 below) that appears in the Equation View 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.
[Note: 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)
the quantum defect. In this exercise we will
confirm the quantum defect level structure for
p-waves in sodium.

The p-wave (l = 1) quantum
defect for sodium is D(l) = 0.86.
Thus, the quantum defect level structure
predicts E_{21} = –0.769 Ry and
E_{31} = –0.218 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 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 Equation View 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 eliminate the discontinuity.
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, where the screening parameter is recorded as b
= 0. Right click anywhere in the equation field for b and select "Edit
parameter.." from the popup menu. 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.] The correct screening value
for sodium p-waves has been found when
no discontinuity is evident in the radial
waveform. Finish by selecting "OK" to
end the edit session with the current settings. For best results, reset the match
point for this waveform --see Technical Note
#2 below. Right click in the equation field for E_{0} and select "Edit
parameter.." from the popup menu., then click the reset
button (but
don't change the energy!). Select "OK", and repeat the adjustment for
b. Finally, restore the
graph to its rightful place by clicking the close button in the upper
right corner of the graph window.

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 the
"OK" button. Notice that this state has the
energy E_{31} calculated
previously (return to Equation View 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 location of the 2p
electron in sodium is that value of r that
maximizes |g_{0}(r)|. Locate
this distance by right-clicking anywhere on the
graph, selecting "Trace" from the popup menu,
and surveying the results.

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
parameter 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.