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PH301
Quantum Computer Assignment
The program for quantum calculations is made by the Consortium for Upper-level Physics Software.
Double-click on the CUPS icon on any of the computers in the Swenson computer lab that runs Windows
95. The software starts up with a list of choices, choose “Quantum Mechanics.”
Part I – Uncertainty Principle
The quantum mechanics program starts up in the "Uncertainty Principle" section, and displays a Gaussian
or "minimum uncertainty" wavepacket. Only the probability densities for the spatial (on the left) and
momentum (on the right) state wave functions are shown. Note that a convenient (if unconventional) set of
units is used. Distances are in nm, and momenta are in units of /nm.
Change the slider control for Delta p to four different values: 0.1, 0.5, 2.0, 6.0 (or as close as you can get).
The program calculates a new Gaussian wavefunction that has that value of momentum uncertainty. It will
recalculate the uncertainty in position and reposition the Delta x slider accordingly. After each adjustment
of the Delta p slider, record the value of Delta p and Delta x. Calculate the product of Delta p and Delta x
for each case.
Now we'll try a different shape for the wavefunction. Choose "Wavefunction Parameters" from the
"Parameters" menu. Select "Triangle," and click OK. Record the value of Delta p and Delta x.
Calculate the product of Delta p and Delta x for this case.
To within roundoff errors, does it seem that the uncertainty principle holds? Why do you think the Gaussian
shape wavefunction is called a "minimum uncertainty" wavefunction?
Part II – Stationary States
Choose “Bound Particles in a Well” from “Section” menu. The program begins with the finite square
well potential. Let’s make it do a more complicated example: the harmonic oscillator. To do this, choose
“Well Parameters” from the “Parameters” menu. Select “User Defined Well,” and define the well as
V(x)=150*x^2 (like a spring: V=1/2 kx2) with a domain from -1 to 1. Click OK. Press F3 to begin the
attempt to find the wavefunction for the ground state of the harmonic oscillator. Sketch the computer’s first
attempt (the white curve). Why can’t this function be a real wavefunction? Press F3 repeatedly until the
wavefunction appears to fit all the requirements of a real wavefunction. How many steps does it take?
Continue pressing F3 until the computer program is also satisfied that it fits all requirements – that is until it
displays the energy of the ground state in the box on the right, and starts over with another impossible
wavefunction in an attempt to find the second energy level wavefunction. Now press F2, which makes it
run quickly through the process of finding wavefunctions. Once it has calculated all the wavefunctions, it
will display them on the right, with the ground state at the bottom.
Now let’s look at how a wavefunction will vary with time. Choose “Wavefunction Parameters” from the
“Parameters” menu. Make a wavefunction that is a combination of the first three energy levels, by setting
component 1 to n=1 and A=0.33, component 2 to n=2 and A=0.33, and component 3 to n=3 and A=0.33.
Note that since this wavefunction is made up of three different energy levels (with equal amplitude of 0.33),
it does not have a definite energy. Click OK. The wavefunction at t=0 is displayed on the left (the real part
in blue, the imaginary part in green). Pay particular attention to the probability density graphed in purple,
because this is what can be determined from experiment. To see how the probability density changes with
time, press F2 -- run. Describe the behavior of the probability density function. Press F2 to stop.
Now make a wavefunction that has a definite energy, E1, by choosing “Wavefunction Parameters,” and
setting component 1 to n=1 and A=1, component 2 to n=2 and A=0, and component 3 to n=3 and A=0.
Click OK. Press F2 -- run. Describe the behavior of the probability density function. Press F2 to stop.
Now make a wavefunction that has a definite energy, E2, by choosing “Wavefunction Parameters,” and
setting component 1 to n=1 and A=0, component 2 to n=2 and A=1, and component 3 to n=3 and A=0.
Click OK. Press F2 -- run. Describe the behavior of the probability density function. Why do you
suppose a state with definite energy is called a “stationary state”?
When you are finished, choose “Exit Program” from the “File” menu. Then choose “Exit Program” from
the main menu. You will be returned to the DOS screen. Type “exit” at the DOS prompt to return to
Windows 95.