Transcript Constructing Models in Quantum Mechanics

Constructing Models in Quantum Mechanics:
Potential Energy Diagrams
Sam McKagan
JILA, University of Colorado at Boulder
Claim: The potential energy diagrams most commonly used in quantum mechanics, such as infinite square wells and square barriers, are abstract
Claim:
models based on extremely sophisticated reasoning and approximations. Standard instruction teaches students to manipulate these models, but
does not address how to build the models or what they represent. Thus, many of our students completely fail to make any connection between
potential energy diagrams and the potential energy involved in real physical situations. Since these potential energy diagrams are a basic
component of nearly all of quantum mechanics, these students essentially have no idea what they’re doing in quantum mechanics class.
They’re not getting it:
The tools we give them:
Traditional Modern Physics Course1:
Students fail to make sense of PE diagrams:
Student Interviews: electron going through square barrier:
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.
Interviewer: 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…
Reformed Modern Physics Course2:
(Designed to address known student difficulties & make models explicit)
Students still struggle to make sense of PE diagrams:
Student Questions:
Clicker Question:
Representations of Potential Energy in
Introductory Physics:
Very familiar:
Less familiar:
U  mgh
GMm
U
r
1 2
U  kx
2
kq1q2
U
r
U  qV
b. Hook up a 5 V battery. Draw a new curve.
All correspond to concrete physical systems!
They come from somewhere!
Representations of Potential Energy in
Quantum Mechanics3:
“the potential”
• We use simplified potentials because real systems
are usually too hard to model.
 



(
x
)

V
(
x
)
(
x
)

E
(
x
)
2
2
m

x
• Determining an approximate potential for a real
system requires knowing what you can ignore.
How do you determine the potential energy
function for a given physical system?
Example: Scanning Tunneling Microscope:
22
Infinite Square Well
Finite Square Well
0|x
|
a
/2
0|x
|
a
/2


V
(x
)
V
(x
)



|x
|
a
/2
V
|
a
/2

0|x

Harmonic Oscillator
1 22
V(x) m
x
2
Hydrogen Atom
2
ke
V(r)
r
“step potential”
“tunneling”
All abstract mathematical constructs!
No relation to real physical systems.
40% draw correct curve:
In Intro, rarely draw diagrams of PE functions.
In QM, rarely talk about physical systems.
 No connection between the two courses.
Implications for Teaching:
• “The potential” in the Schrodinger equation refers
to the potential energy of a particle as a function of
position.
• These simplified potentials can sometimes be good
approximations of real systems.
“free particle” V=0
40% draw correct curve:
• In QM, we use potential energy instead of forces to
describe interactions between objects.
• This potential describes the interactions of the
particle with its environment.
Student Responses:
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.
What do experts know that we never talk about?
How to help students build
models of PE diagrams:
Minimum:
• Practice determining
We must be more explicit about the
potentials for real
models we are using in QM so students
physical situations
have some idea what they’re doing.
and vice versa.
Ideal:
• Analyze underlying
assumptions
Teaching QM is a great opportunity to
. • …
teach students how to build models!
Example: modeling electron in
wire as finite square well
V(x)
SAMPLE
METAL
Tip
V
I
I
• “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?”
Unspoken assumptions:
Sample
Tip
• 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.
• The potential difference between the tip and the
sample tells you the potential difference between
two points just outside the metals, not inside.
• You can ignore collisions of the electron with
other electrons and atoms.
End Notes:
References:
1. S. B. McKagan and C. E. Wieman, 2005 PERC Proceedings (2006).
2. S. B. McKagan, K. K. Perkins, and C. E. Wieman, 2006 PERC
Proceedings (2007).
3. Pictures of potential energy representations are taken from the Physics
Education Technology Project simulations Quantum Bound States and
Quantum Tunneling, available at: http://phet.colorado.edu.
Acknowledgements:
Thanks to the NSF for providing the support for this project, and to all
the members of the PhET Team and the Physics Education Research at
Colorado group (PER@C).