The applet models a mass m (the bouncer) in a
uniform gravitational field directed along the
x-axis, and described by the potential energy
V(x) = mgx for x
> 0. A virtually impenetrable surface at x = 0
marks the position of the 'floor'. The choice of 'Flex units'
here allows for the incredibly small scales of energy and
distance needed to describe the low-lying states of a
macroscopic bouncer, like a tennis ball. In these units, the
scales for distance (d) and energy (E)
are related as E·d2 =
ћ2/2m. We take d
= 10−22 m to give an energy unit E ~
10−23 J for m = 50 g. The gravity force
on the bouncer F = mg is the slope of the linear
segment of the potential curve, about 0.5 J/m, or 5 in Flex
units. This value for F appears in the
Equation View of the applet, along with
U, the minimum energy the bouncer needs to
penetrate the floor. The default U = 10000
(~10−19 J) is unrealistically small, but still
large enough to accurately model the low-lying states of the
bouncer. On the tab labeled Graphics: [x] the
bouncer potential energy function V(x)
is plotted over the interval [–1, +10]. The listing to the
right of the graph includes placeholders for three stationary
states of the bouncer. In this exercise we will find the
three lowest stationary states of the bouncer and confirm
that energy is quantized even for macroscopic objects like a
bouncing tennis ball.
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
bouncer.
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; 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
bouncer. Count the number of nodes for the wavefunction to
see which stationary state you have found. We seek the
ground state (nodeless), first excited state (one node), or
the second excited state (two nodes). 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 and third
placeholders in the list, using them to find the remaining
stationary states, i.e., those not found in the preceding
step. Recall that the stationary state energies you have
found are in units of 10−23 J!
For each stationary state, there is a most probable height
from the floor for the bouncer, identified as that value of
x for which the probability amplitude
|ψ(x)| is largest. Locate this position for each
state by right-clicking anywhere on the graph, selecting
"Trace" from the popup menu, and surveying the results.
Remember that all numerical values for x are in
units of 10−22 m! Compare the quantum
predictions with their classical counterparts, discussing
any discrepancies in the context of Bohr's Correspondence
Principle. [Classically, the bouncer is most likely to be
found at the top of its flight, where the speed drops to
zero.]
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: The
Quantum Bouncer
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
bouncer.
re-positions the match point to an extremum of the current
waveform.
