Exploratory Lab on Basic Wavefunction Properties
CH351 Physical Chemistry Dry Lab I (Prof. Wu)
Introduction: We will use an applet that simulates quantum mechanical wavefunctions
for a particle inside a potential well. This applet allows changing various parameters of
the system, such as the shape, depth and width of the potential well, particle mass. One
can examine several possible stationary state wavefunctions and superpositions of
stationary states. The applet can display the probability density as well as the
wavefunction, including real and imaginary parts, magnitude and phase (encoded as
color).
Objective: To gain “hands-on” experience with wavefunctions to make them less
abstract and to help develop intuition on how wavefunctions behave. Specific objectives
include 1) recognizing analogies with classical wave behavior, 2) identifying trends of
how the stationary states behave with varying energy and system parameters, and 3)
observing time-dependent behavior associated with superpositions of stationary states.
1. Load the applet and familiarize yourself with the controls.
The applet can be launched by first visiting the PhET website at
http://phet.colorado.edu/web-pages/simulations-base.html In the category “Quantum
Phenomena”, click on the applet “Quantum Bound States”. The applet should load in a
separate window.
The applet window will show three tabs at the top: for this lab, we’ll just be using the
first tab “One Well”, which simulates a particle in a single potential well. A potential
well refers to the situation where the potential energy is lower in one region in space.
There are several functional forms for the potential energy available, selectable by the
“Energy chart” panel on the right.
The main graph in the top left shows the potential energy well (purple line), as well as the
energies of the stationary states (green horizontal lines). Recall that the stationary states
are the special states that are analogous to standing waves in classical systems (e.g. waves
on a string). You can click on the green lines and see the corresponding wavefunction or
probability density plotted in the panel below, depending on what you choose to be
displayed in the panel on the right labeled “Bottom chart”. Note that the phase, between
0 and 2π, is displayed as a color, as indicated in the panel. The last panel on the bottom
right controls the particle mass in units of an electron mass.
Play around and familiarize yourself with the controls. You can adjust the speed of the
animation by moving the slider at the bottom between normal and fast.
2. Wavefunction and energy trends.
Set the “Potential Well” tab to be “square”: this corresponds to a “square well potential”,
which has a finite region of constant lower potential. Vary the various parameters
available to you, and report any trends you find, including:
a) How does the ground state (lowest energy state) energy change with well width,
depth and particle mass? Explain in terms of the Heisenberg uncertainty
principle.
b) How does the shape of the stationary state wavefunctions vary with increasing
energy?
c) How does the frequency of oscillation of the stationary state wavefunctions vary
with increasing energy? Specifically, for the first few stationary states, record the
frequency by clocking the time for one period. What do you think is the
mathematical relationship between frequency and energy?
d) What’s the relationship between the time dependence of the real and imaginary
parts of the wavefunction, and the magnitude of the wavefunction?
3. Superposition states and time dependence.
As was the case for standing waves in a string, we see that stationary states have a spatial
envelope (the magnitude) that doesn’t vary in time, even though the wavefunction itself
oscillates. As a result, the probability density will also be unchanging in time for
stationary states. Verify this.
However, as we saw in an example in class, two or more stationary states can be added to
form a new state, called a superposition state, which does not have a fixed envelope
function. This is analogous to the superposition states you did in the Mathematica
portion of the dry lab for vibrations on a string. When the components of the
superposition state have different oscillation frequencies, the different regions that
interfere constructively and destructively will shift in time.
Create a superposition state by clicking on “Superposition state…” What are shown are
the coefficients ci multiplying stationary state wavefunction i in the sum of
wavefunctions. Change the value of c2 to be 1.00. This creates a superposition state with
equal contributions of the ground state and the first excited state. This function is not
normalized, and so clicking on “Normalize” will multiply all the coefficients by a
constant so that the total wavefunction will be normalized. Then click “Apply” to display
the function.
a) Clock the time it takes for the probability density of this superposition state to
make a full cycle. Clock the time it takes for the wavefunction to make a full
cycle (come back to its original magnitude and phase). Are these two times the
same? Why or why not? Compare these times to the period of the ground and
first excited state (as you measured in question 2c above). What is the
relationship between the two?
b) Try other linear combinations (sums) to form other superposition states. Record
any observations you notice.
4. Explore other potentials.
The other options for “Potential Well” correspond to other chemically relevant situations.
The asymmetric potential corresponds to a confined particle that feels a constant force
(why?), for instance an electron in a semiconductor subject to an electric field. The 3d
Coulomb potential corresponds to the hydrogen atom s-orbitals. The harmonic oscillator
corresponds to vibrations of a diatomic molecule. Pick one and explore it. Write a short
paragraph on what you observed. Some suggested questions for these systems:
a) How do the number of nodes vary with energy?
b) How does the frequency of oscillation compare to the case of the square well (is it
the same order of magnitude)?
c) How smoothly do the wavefunctions go to zero?
d) How does the spacing of the quantized energies (energies of the stationary states)
compare to the square well?