Student Understanding of Potential Energy Diagrams in

Quantum MechanicsSam McKaganPER seminar

Colorado School of MinesOctober 30, 2007

Acknowledgements

Faculty:Faculty: Michael DubsonMichael Dubson Noah FinkelsteinNoah Finkelstein Valerie OteroValerie Otero Kathy PerkinsKathy Perkins Steven PollockSteven Pollock Carl Wieman Carl Wieman

Programmers:Programmers: Chris MalleyChris Malley Sam ReidSam Reid Ron LemasterRon Lemaster John deGoesJohn deGoes

Postdocs: Postdocs: Stephanie ChasteenStephanie ChasteenLaurel MayhewLaurel MayhewSam McKaganSam McKaganArchie PaulsonArchie Paulson

Grad Students:Grad Students:Wendy AdamsWendy AdamsCharlie BailyCharlie BailyMariel DesrocheMariel DesrocheKara GrayKara GrayLauren KostLauren KostNoah PodolefskyNoah PodolefskyChandra TurpenChandra Turpen

2/||

2/||0)(

ax

axxV

2/||

2/||0)(

0 axV

axxV

r

kerV

2

)(

22

2

1)( xmxV

“free particle” V=0

“step potential”

Infinite Square Well Finite Square Well

Harmonic Oscillator Hydrogen Atom

“tunneling”

Potential Energy Diagrams

)()()()(2 2

22

xExxVxxm

“the potential”

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)

• Interviews with 4 students at end of traditional MP.• All regularly attended optional weekly help sessions.• Average to

above averagestudents:

• 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

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

F D- D D+ C- C C+ B- B B+ A- A

Course Grade Distribution

Experimental Evidence 1: they don’t get it

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.

Student 1: Course grade B+

Experimental Evidence 1: they don’t get it

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…

Student 2: Course grade A-

Representations of Potential Energy in Introductory Physics:

mghU r

qkqU 21

r

GMmU qVU 2

2

1kxU

All correspond to concrete physical systems!They come from somewhere!

Familiar:

Less Familiar:

Theoretical Argument 1: why they don’t get it

Representations of Potential Energy in Quantum Mechanics:

)()()()(2 2

22

xExxVxxm

“the potential”

Theoretical Argument 1: why they don’t get it

2/||

2/||0)(

ax

axxV

2/||

2/||0)(

0 axV

axxV

r

kerV

2

)(

22

2

1)( xmxV

“free particle” V=0

“step potential”

All abstract mathematical

constructs!No relation to real physical systems.

Infinite Square Well Finite Square Well

Harmonic Oscillator Hydrogen Atom

“tunneling”

Representations of Potential Energy in Quantum Mechanics:

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.

Theoretical Argument 1: why they don’t get it

Part 2

Experimental Evidence and Theoretical Argument from

reformed modern physics course (2130 – for engineers)

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.

Experimental Evidence 2: student questions

Short copper wire, length L.

What is V(x)?

Consult with group.Will call on random groups for ideas.

0 L

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’smove aroundto lowest PE

repel other electrons = potential energy near that spot higher.as more electrons fill in, potential energy for later ones getsflatter 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 getsflatter and flatter. For top ones, is VERY flat.

+ + + + + + + + + + + + + + + +

How could you find out how deep the pit is for the topelectrons in copper wire?

PE for electrons with most PE. “On top”

How could you find out how deep the pit is for the topelectrons in copper wire?

This is just the energy needed to remove them from the metal.That is the work function!!

as more electrons fill in, potential energy for later ones getsflatter and flatter. For top ones, is VERY flat.

work function ofcopper = 4.7 eV

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?”

• 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

What do experts know about PE diagrams that we never talk about?

SampleTip

I

Tip

V

I

SAMPLE METAL

V(x)

How do you determine the potential energy function for a given physical system?

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

Example: Scanning Tunneling Microscope

• “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?”

Theoretical Argument 2: unspoken assumptions

STM questions from students

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.

b. Hook up a 5 V battery. Draw a new curve.

40% draw correct curve:

40% draw correct curve:

Reformed instruction helps, but not enough

Student Responses:

Clicker 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 c. I=false, II=true b. I=true, II=false d. I=false, II=false

Exam Question:

58% correct18% 18%7%

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…?