The applet simulates a proton confined to the nucleus of an
atom. The nuclear potential is modeled here as a square well
with height U = 26.0 MeV and width L =
10.0 fm (1 fm = 10−15 m = 10−6 nm).
These values appear in the Equation View of the
applet, along with the proton mass = 938.38
MeV/c2. On the Graphics:
[x] tab the square well potential energy
V(x) is plotted over the interval [–10
fm, +20 fm]. The listing to the right of the graph includes
placeholders for two stationary states of the proton in this
well. In this exercise we will find the two lowest stationary
states of this nucleon and confirm the notion of energy
quantization for this case.
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
proton.
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 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 point by typing directly into the text field.] The
button at the far right resets the matching point, and
should be used when nearing a correct energy – see
Technical Notes below. When no discontinuity is evident,
the energy is "allowed" and the wavefunction is one of the
stationary states for the proton in this well. Count the
number of nodes for the wavefunction to see which
stationary state you have found. We want the ground state
(nodeless) or the first excited state (one node). If you
have found another, continue searching at lower energies.
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, using it to find the other stationary state
(first excited state, or the ground state), i.e., the one
not found in the preceding step.
The Numerov method is used to construct 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: Proton
in an Atomic Nucleus
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
proton.
re-positions the match point to an extremum of the current
waveform.
