Student Understanding of
Potential Energy Diagrams in
Quantum Mechanics
Sam McKagan
PER seminar
Colorado School of Mines
October 30, 2007
Acknowledgements
Faculty:
Michael Dubson
Noah Finkelstein
Valerie Otero
Kathy Perkins
Steven Pollock
Carl Wieman
Programmers:
Chris Malley
Sam Reid
Ron Lemaster
John deGoes
Postdocs:
Stephanie Chasteen
Laurel Mayhew
Sam McKagan
Archie Paulson
Grad Students:
Wendy Adams
Charlie Baily
Mariel Desroche
Kara Gray
Lauren Kost
Noah Podolefsky
Chandra Turpen
Potential Energy Diagrams
“the potential”
 2 2
 ( x)  V ( x) ( x)  E ( x)
2
2m x
Infinite Square Well
0|x
|
a
/2

V
(x
)


|x
|
a
/2

Harmonic Oscillator
1 22
V(x) m
x
2
“free particle” V=0
“step potential”
“tunneling”
Finite Square Well
0|x
|
a
/2

V
(x
)

V
|
a
/2
0|x

Hydrogen Atom
2
ke
V(r)
r
Results
• After completing typical course in MP or QM,
many students have no idea what potential
energy diagrams mean.
• Relating PE diagrams to physical system
requires sophisticated modeling.
• Instruction that explicitly addresses how to build
models of PE leads to really deep student
questions.
– Shows how hard this topic is.
– Makes you wonder about classes where they don’t
ask these questions.
• Even in reformed courses, students still have
problems.
Outline
• Part 1 – traditional modern physics class
– Experimental evidence
– Theoretical argument
• Part 2 – reformed modern physics class
– Experimental evidence
– Theoretical argument
• Part 3 – remaining problems and
what to do next
Part 1
Experimental Evidence and
Theoretical Argument from
traditional modern physics course
(2130 – for engineers)
Experimental Evidence 1: they don’t get it
• Interviews with 4 students at end of traditional MP.
• All regularly attended optional weekly help sessions.
Course Grade Distribution
• Average to
20%
above average 18%
16%
14%
students:
12%
10%
8%
6%
4%
2%
0%
F
D-
D D+ C-
C C+ B-
B
B+ A-
A
• All 4 claimed that PE diagram did not represent
potential energy of electron, even though it was
labeled as such…
Experimental Evidence 1: they don’t get it
Student 1:
Course grade B+
Interviewer: If this curve that you drew is the potential
energy, then what is this square thing that’s drawn here?
Student 1: I don’t know, that’s just the bump that it goes
through. I don’t know what it means. I just see that and I
know that it’s some kind of obstacle that it goes through.
Experimental Evidence 1: they don’t get it
Student 2:
Course grade AInterviewer: What does the potential energy looks like for this case?
Student 2: For the electron? I guess it would be a straight line here, and
then… well, it would have a certain potential energy, wouldn’t it? Going
up to the gap? I’m not exactly sure. I don’t know what it would… I don’t
know what the potential energy for the electron would look like.
Interviewer: So this thing that’s being plotted here, U(x), what is that?
Student 2: Potential energy. I guess it’s the potential energy of the, I’m
not exactly sure. I know that the barrier, within the barrier, the
potential energy increases. So I guess it would be a measure of the
potential energy of the medium that it’s in, of some sort, I’m not exactly
sure.
Interviewer: But it’s not the potential energy of the electron?
Student 2: Um, I don’t, not, that doesn’t ring a bell to me, why it would be.
That doesn’t come to my mind. I don’t know, I guess it could be, but…
Theoretical Argument 1: why they don’t get it
Representations of Potential Energy in Introductory Physics:
kq1q2
U  mgh
U
r
Familiar:
GMm
U
r
U  qV
1 2
U  kx
2
Less
Familiar:
All correspond to concrete physical systems!
They come from somewhere!
Theoretical Argument 1: why they don’t get it
Representations of Potential Energy in Quantum Mechanics:
“the potential”
 
 ( x)  V ( x) ( x)  E ( x)
2
2m x
2
2
Theoretical Argument 1: why they don’t get it
Representations of Potential Energy in Quantum Mechanics:
Infinite Square Well
0|x
|
a
/2

V
(x
)


|x
|
a
/2

Harmonic Oscillator
1 22
V(x) m
x
2
“free particle” V=0
“step potential”
“tunneling”
Finite Square Well
0|x
|
a
/2

V
(x
)

V
|
a
/2
0|x

Hydrogen Atom
2
ke
V(r)
r
All abstract
mathematical
constructs!
No relation to real
physical systems.
Theoretical Argument 1: why they don’t get it
In Intro, rarely draw diagrams of PE functions.
In QM, rarely talk about physical systems.
 No connection between the two courses.
Part 2
Experimental Evidence and
Theoretical Argument from
reformed modern physics course
(2130 – for engineers)
Experimental Evidence 2: student questions
Reformed Curriculum
• Context: Every PE diagram given in terms of
physical system. (e.g. square wells = electrons
in wires)
• Building models: interactive lectures, homework
problems, and a tutorial in which students build
up PE diagrams for square wells, STM, alpha
decay, & more.
• Addressing student difficulties: clicker questions
confront belief that vertical axis of PE diagram =
height, confusion between PE and TE, etc.
0
L
Short copper wire, length L.
What is V(x)?
Consult with group.
Will call on random groups for ideas.
Remember photoelectric effect.
Took energy to kick electron out. So wants to be inside wire.
 inside is lower PE.
Everywhere inside the same?
PE
+
1 atom
+
+
+
+
+
+
+
+
+
many atoms
but lot of e’s
move around
to lowest PE
repel other electrons = potential energy near that spot higher.
as more electrons fill in, potential energy for later ones gets
flatter and flatter. For top ones, is VERY flat.
PE for electrons with most PE. “On top”
+ + + + + + + + + + + + + + ++
as more electrons fill in, potential energy for later ones gets
flatter and flatter. For top ones, is VERY flat.
How could you find out how deep the pit is for the top
electrons in copper wire?
PE for electrons with most PE. “On top”
work function of
copper = 4.7 eV
as more electrons fill in, potential energy for later ones gets
flatter and flatter. For top ones, is VERY flat.
How could you find out how deep the pit is for the top
electrons in copper wire?
This is just the energy needed to remove them from the metal.
That is the work function!!
Experimental Evidence 2: student questions
•“I have trouble understanding what the potential is when we are
looking at models of an electron in a wire, free space, finite square
well, infinite square well. I am sort of getting this idea of it being
similar to a work function in that once the potential (V) is less than
the potential energy, the electron is out of the wire. I can usually
follow the math/calc that follows the examples okay, but the
overall concept of this potential (V) still confuses me, and so I still
don't have a firm grasp of [what] the square well models
mean/represent/whatever.”
•“I cant find a general description of an infinite well, i understand
what it does but not what it is or where its used.“
•“Voltage is used when we talk about electromagnetic forces, like
the coulomb force. What I'm confused about is that we used a
voltage well to show the strong force in effect. Is it accurate to
show the strong force as a very deep voltage well?”
Theoretical Argument 2: unspoken assumptions
What do experts know about PE diagrams
that we never talk about?
• In QM, we use potential energy instead of forces to
describe interactions between objects.
• “The potential” in the Schrodinger equation refers to the
potential energy of a particle as a function of position.
• This potential describes the interactions of the particle
with its environment.
• We use simplified potentials because real systems are
usually too hard to model.
• These simplified potentials can sometimes be good
approximations of real systems.
• Determining an approximate potential for a real system
requires knowing what you can ignore.
Theoretical Argument 2: unspoken assumptions
How do you determine the potential energy
function for a given physical system?
Example: Scanning Tunneling Microscope
V(x)
SAMPLE
METAL
Tip
V
I
I
Sample
Tip
Theoretical Argument 2: unspoken assumptions
Example: Scanning Tunneling Microscope
• Potential is uniform inside a conductor, so V(x) is flat in tip and
sample (only works if sample is conductor).
• Complete circuit in steady state, so electron flow doesn’t change
potential.
• Because an electron is bound to a metal, it has a different potential
energy in the metal than in the surrounding air. The difference
between these two potential energies is given by the work function
of the metal.
• To analyze this system, we need to look at the potential energy of
any one electron due to its interactions with all the other atoms and
electrons in the metal of both the tip and the sample, and with the
electric field of the applied voltage.
• If there is a voltage across a region of space, the potential energy of
an electron in that region is a linear function of position.
• Potential difference between the tip and the sample is the potential
difference between two points just outside the metals, not inside.
• You can ignore collisions of the electron with other electrons and
atoms.
Theoretical Argument 2: unspoken assumptions
STM questions from students
• “As the electrons tunnel through, isn’t the sample
potential energy going to drop?”
• “The quantum tunneling microscope can be used on any
material even though not every material has a “sea” of
electrons? Wouldn’t losing an electron in a crucial
covalent bond break the molecule apart?”
• “Can we really follow the behavior of a single electron?”
• “What about interactions with atoms?”
Part 3
Remaining Problems
and what to do next
Reformed instruction helps, but not enough
Exam Question: An electron is tunneling from a
scanning tunneling microscope (STM) tip to
sample’s surface. The tip’s work function is 4 eV
and the sample’s work function is 5 eV .
a. Draw potential energy curve if no
voltage between tip and surface.
40% draw correct curve:
b. Hook up a 5 V battery.
a new curve.
40% draw correct curve:
Draw
Reformed instruction helps, but not enough
Clicker Question:
Student Responses:
Exam Question:
For an electron in the n=2 state, which of the following statements are true:
I. The potential energy of the electron is greater than 0.
II. The potential energy of the electron is greater than the potential energy
of an electron in the n=1 state.
a. I=true, II=true 18% c. I=false, II=true 18%
b. I=true, II=false 7% d. I=false, II=false 58% correct
Conclusions
• Many students have no idea what potential
energy diagrams mean.
• We’ve figured out how to get them asking
the right questions and learn a bit more.
• How to improve learning further…?