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Modeling Molecules
p. 165
Intuitive Quantum Physics
Name:___________________________
Last lab, you used a particle in a single square well to model a single atom. This would be great if all of
nature was single atoms floating around, but atoms tend to get together and form molecules. It’s
relatively easy to model molecules made of two atoms of the same element, bound together by a covalent
bond in which the atoms share their outer electrons. A common molecule of this type is O2, the type of
oxygen that naturally appears in the atmosphere. It’s what we breathe.
How would you go about modeling a molecule of two atoms, close together? The goal of today’s lab is to
use our toolbox to do just that.
I.
The Two Potential Well System or a Covalent Molecule
Consider a potential energy well width of x = 2 and a PE of –30 eV. This square well is drawn on the
PE(x) graph below. Label this well R, since it will be the right well (brilliant! what a great name!).
Suppose there is an electron confined to this well.
A. On the (x) graph, sketch the ground state wave function, keeping in mind what types of functions
are allowed inside and outside of the well. On the |(x)|2 graph, sketch the probability density.
PE(x)
40
30
20
10
0
-8
-7
-6
-5
-4
-3
-2
1.
5
0
1
2
x
3
0
1
2
x
3
0
1
2
x
3
-10
-20
-30
-40
(x)
(x)
-8
-7
-6
-5
-4
-3
-2
-1
|(x)|2
-8
-7
-6
-5
-4
-3
-2
 2003-6, University of Maine Physics Education Research Laboratory.
-1
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Modeling Molecules
B. Suppose an identical potential energy well (width of x = 2 and a PE of –30 eV) is added to this
system, but located a very large distance to the left of the initial well on the PE(x) graph. This well is
labeled L. (Any guesses why? Brilliant, that’s us!)
When the two wells are very far apart, three things are true:

The probability density graph for the electron does not change.

The amplitude of the wave function for the electron doesn’t change.

The curviness of the wave function doesn’t change.
1. What does this tell us about the probability of finding the electron in well R? In well L?
2. Did we need to tell you the second and third bullet, or was the first one actually sufficient?
Explain.
II. Board Meeting 1
Answer all the questions in your group and prepare a board with any sketches or comments you need to
support your ideas.
Imagine you move well L closer to well R. What do you think will happen to:
1. The energy of the ground state?
2. The wave function ?
3. The probability of finding the electron in well L?
Modeling Molecules
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III. Dealing with problematic computer representations
Launch Safari and navigate to:
http://www.zaposa.com/two-well.php
(This should be bookmarked on your Safari browser.)
Note that two-well.php is quite different than the square well you used earlier. The text box labeled
"Energy" will tell you the energy of the electron. You can’t type in it, though. The arrow buttons will let
you change the energy in small steps, just as before. At the bottom of the screen, you will find two text
boxes. The first will let you change the depth of the two wells. The second will let you change the
spacing between the wells. In another text box, you can also type in the energy level that you want.
Please, only do what we tell you to do. The first time you use the software, it’s important to develop your
ideas in the order that we ask you to. Later, at home, you can play around with it more. Okay? Thanks.
A. The program plots the wave function for this scenario for the values of TE that you enter.
Unfortunately, rather than showing the energy in one graph and  in another, it overlaps the two,
showing only one line. We happen to hate this overlapping of graphs. Let’s deal with that first.
1. What is the total energy of the particle at x = 0 when the page first loads? What part of the
software are you looking at to answer the question? (This may seem so blindingly obvious that
you think we’re stupid in asking, but we have a reason for asking!)
2. What is the total energy of the particle at x = 0.75 when the page first loads? How can you tell?
(What part of the software did you use to answer the question – the same as before?)
3. The energy text box reports only one energy for the entire graph. What does this fact imply about
the total energy of the particle for every value of x?
4. The graph uses energy as its vertical axis.
a. In which ways is that correct? Which color line is correctly graphed in this fashion?
b. In which ways is the vertical axis label misleading? Refer specifically to the wave function.
Many authors use the dual representation in text and popular science books, even though it’s not very
good -- it’s your responsibility to recognize that two quantities are being shown on one graph!
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Modeling Molecules
B. We have to deal with another crazy element of the visual representation before we move on.
1. Where does the x-axis for the wave function seem to be? How do you know?
2. Is the electron at a physically allowable energy? If so, which state is it in? If not, explain how
you know. You won’t be able to answer this question correctly if you haven’t answered #1…
3. Where is the particle most likely to be? Give a number, x = …
4. What is the probability that the particle is in the left well?
IV. Building a molecule made up of two atoms
A. Now that we’ve worked out how to interpret wave functions and energy graphs drawn on the same
diagram, let’s look at what they actually mean. Fill in the following table (you already did the first
column, so copy over your results and then fill in the other two columns):
Describe changes to the…
1.4 nm
when the well spacing is…
1.1 nm
0.6 nm
energy of the electron
(give its value…)
energy state of the electron
(give the number…)
most likely location
of the electron
(give the location…)
probability of finding the
electron in the left well
(explain your interpretation)
Tell the story of an electron attached to atom R as you move atom L closer and closer to R.
(This might be a good time to recognize and debate the crazy things that happen in quantum physics.)
Modeling Molecules
B. Change the well spacing to 0.1 nm.
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PE (x)
1. What is the energy of the electron?
x
2. Compare and contrast the shape of the wave function for this
state to the wave function in the ground state of a single well
that extends from the left side of L to the right side of R.
(How is it similar? How is it different?)
PE (x)
x
3. Where is the particle most likely to be for this arrangement?
Compare it to being in a single wide well…
4. One interpretation of this wave function is that an electron which is located in one well is always
attracted to the other well. How can you support this interpretation? Be as specific as possible
about what you know about electrons being attracted to things and how the wave function plays a
role in your answer.
C. Change the well spacing to 0.01 nm. Compare and contrast this state to that of a single, wide well.
1.
Where are you most likely to find the electron?
2. How, if at all, did the curviness of wave function at a given value of the wave function in the
wells change as the spacing got smaller? You may find it helpful to change the spacing to 1.4 nm
again.
3. As the two wells are moved closer together, what happens to the energy for the ground state?
4. Describe how your answers for 2 and 3 agree with the Schrödinger equation,
Curv x k  TE particle  PEsystem x

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Modeling Molecules
D. Think about two atoms very close together, sharing an electron.
1. When two atoms are close together, where are you most likely to find the electron?
2. Would you always find the electron where you described it in question 1?
(If you haven’t already had a conversation about the crazy things that happen in quantum physics,
then this would really be a good time to have it – could a cart do what this electron is doing?!)
V. Molecules have excited states, too
Change the depth of the wells to –90 eV, and the spacing to 0.6 nm. A portion of picture is reproduced
below. We have separated the PE graph from the wave function graph. Label the axes on both graphs.
0
-1
-0.5
0
-100
A. Sketch the ground state as given by the computer.
1. Where is the particle most likely to be?
2. What is the energy of the ground state?
0.5
1
1.5
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B. Now, we want to find the first excited state for this system. There's a button labeled "Find" next to a
text box labeled "n". Change n to 2, and click Find. You may need to click find twice (it’s a software
thing). Sketch this state. Label axes…
1. How do you know that this is a physically allowable energy level?
2. Before, we used the “count the humps” rule to help us determine the state of the electron. This is
the first excited state, but it has the same number of humps as the ground state. Something has
changed. What is it?
3. Where is the particle most likely to be?
4. What is the energy of this state?
VI. Board Meeting II
Your instructor will assign your group one of the following questions:
1. In V.B the computer drew a wavefunction for an electron in the first excited state. Is it possible
to sketch another  with this energy? If so, draw it below and on your board. (Label your axes!)
If not, state so explicitly.
2. What is the change in energy between the first excited state and the ground state for an electron in
the configuration of Section V? Can you see the photon created by this energy change? (The
energies of visible light photons range from 1.65 to 3 eV.)
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Modeling Molecules
VII. Energy level splitting
A. Change the depth of the well to –300 eV and the spacing to 10 nm. Notice that only one well shows
up on your computer screen (the other is too far to the left).
1. What is the energy of the ground state for this configuration?
2. What is the energy of the first excited state?
B. Keep the depth at –300 eV and change the spacing to 0.6 nm.
1. What is the energy of the ground state for the new configuration?
2. Imagine you found a particle at x = –0.75 nm, in the first excited state. What energy would it
have? (If you’re having a hard time answering this question or were even momentarily tricked,
then go back and think about graphing two things on one diagram!)
Three students are discussing the double well scenario:
Student 1: So, the wave function in the left-hand well curves down,
while the wave function in the right-hand well curves up. That
means the wave function is an
s-function in the left-hand well, and an e-function in the righthand well, right?
Student 2: Wait a minute – the wave function is higher on the left
than the right, so the electron has more energy in the left-hand
well than the right-hand well. How can this be one energy state?
Student 3: No, the energy is the same everywhere, it just means that
you’re more likely to find the particle in the left-hand well
than in the right-hand well.
3. Pretend that these students are in your group. Discuss their ideas – with what parts do you agree
or disagree? What would you tell these students to help them come to a consensus?
4. Can an electron in this “molecule” have an energy of -274.228 eV? Explain.
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C. Choose n = 3 so that the electron is in the second excited state.
1. What is the energy of the electron in this state?
2. Sketch the second excited state below and mark where it is an e-function or an s-function. Make
sure to label the axes and mark where the edges of the wells appear.
D. Choose n = 4 to get the third excited state.
1. How does it differ from the second excited state (n = 3) in terms of…
a. …energy?
b. …the shape of the wave function?
2. How are the wave functions of the second and third excited states similar?
E. Now look at the fourth and fifth excited states (that’s n = 5 and n = 6…).
1. Compare the fourth and fifth excited states in terms of…
a. …energy
b. …the shape of the wave function
2. Again, you should notice a similarity in the wave functions. What is it?
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Modeling Molecules
F. Now we’re going to think about the energy levels. A bit of a review – how do we graph energy
levels?
G. On the graph below right, graph the energies of all of the bound states.
(Notice that it’s legal to do this, since you’re graphing energies on an energy graph…)
E(x)
What pattern do you notice about the
spacing of the energy levels?
-1
-0.5
-20 0
x
0.5
1
1.5
-70
H. Go back and think about the different
wave functions you’ve looked at when
the atom spacing is 0.6 nm. Think
about the parts of the wave functions
inside the wells.
1. How many humps are inside the
wells for the ground and first
excited states?
-120
-170
-220
Second and third excited states?
Fourth and fifth excited states?
2. Compare your findings from 1 to
the number of humps for the
ground, first, and second excited
states for a single square well.
-270
-320
3. Does this give any support to the title of this section, “Energy Level Splitting”? In other words,
what happens to the energy levels of a single deep well when you add a second deep well? It
might help to go back to VII.A. on page 172 to think about this.
(You might want to know that what you’re describing lets us understand how electrons in wires
are free to move around and carry electricity – yup, we can use quantum physics to describe how
electrons flow in a wire.)