The applet shows the potential energy for an electron
confined to a finite square well of width 0.200 nm and height
100 eV (these values appear in the Equation
View, along with the electron mass = 511
keV/c2). The listing to the right of
the graph includes placeholders for three stationary states
for the electron in this well. In this exercise we will find
the two lowest-lying bound states and determine the total
number of bound states this well can support. The exercise
illustrates energy quantization, i.e., only certain energies
are permitted for the electron, corresponding to states which
satisfy the acceptability criteria for quantum wavefunctions.
Instructions for use
Show the first waveform in the list (labeled ψ0)
by right-clicking its placeholder, clicking the visibility
icon
beside the "Real" label in the Colors | Visibilities
field, then choosing the "OK" button. The displayed
wavefunction has a noticeable discontinuity. Since quantum
wavefunctions must be everywhere continuous, the energy of
this state cannot be one of the allowed energies for the
electron.
Adjust the energy of this state upward from zero 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 | resize the graph as
desired and proceed to Equation View, where
the energy of this state is recorded as
E0 = 0. Right click anywhere in the
equation field 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 button at the lower right resets the
matching point, and should be used when nearing a correct
energy – see Technical Notes below. When no discontinuity
in the waveform is evident, the energy is "allowed" and the
wavefunction is one of the bound states for the electron in
this potential well. Count the number of nodes for the
wavefunction to see which bound state you have found.
Finish by selecting "OK" to end the edit session with the
current settings. Finally, restore the graph to its
rightful place by clicking the close button in the upper
right corner of the graph window.
Repeat the above procedure for the second placeholder in
the list, labeled ψ1. If you found the ground
state (no nodes) in the previous step, search now for an
acceptable wavefunction with the next-lowest energy; this
is the first-excited state, recognized by having exactly
one node. Note the different symmetries for the two states:
the ground state is symmetric about the midpoint of the
potential well, the first-excited state is antisymmetric
about this point.
Repeat the procedure once more for the third placeholder,
labeled ψ2, this time searching for an
acceptable wavefunction with the highest possible energy,
but still "localized" in the well. Count the number of
nodes to discover which excited state you have found.
Adding one to this number (for the ground state) gives the
total number of electron bound states this particular well
can support.
The Numerov method is used to contruct stationary waves for
a given (trial) energy. Using boundary conditions derived
from the correct asymptotic form first at the left
endpoint, and then again at the right 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.
Exercise: The
Finite Square Well
beside the "Real" label in the Colors | Visibilities
field, then choosing the "OK" button. The displayed
wavefunction has a noticeable discontinuity. Since quantum
wavefunctions must be everywhere continuous, the energy of
this state cannot be one of the allowed energies for the
electron.
re-positions the match point to an extremum of the current
waveform.
