Download 179 tut Tunneling - University of Maine Physics Education

Tunneling - a Quantum Model Consequence
Intuitive Quantum Physics
I.
p.179
Name:___________________________
The Bowling Ball Model of Large Objects
About one hundred years ago, the model for light was that it was a wave/particle, but the model for
electrons was that they were only particles. In thinking of electrons as having only particle properties,
one arrives at different predictions for the probability density than if one considers the electron to have
both particle and wave properties. And that leads to some unexpected consequences…
Recall from previous tutorials that kinetic energy and gravitational potential energies have been defined
as
KE  12 mv 2
PE  mgh
We can use the idea that the total energy (the sum of the kinetic and gravitational potential energies) is
conserved in many situations to predict the behavior of systems, that is, TE = KE + PE = constant.

A bowling ball is in a two-tiered system, shown below. If we define the lower level to be at height zero,
the bowling ball has a gravitational potential energy of 0 J. The height of the upper level is such that the
gravitational potential energy of the bowling ball on that level is 30 J.
PE = 30 J
PE = 0 J
h
A. Sketch a picture graph of the gravitational potential energy of the bowling ball as a function of
position. Label your axes.
 2003-6, University of Maine Physics Education Research Laboratory. (200701)
Tunneling – a Quantum Model Consequence
p.180
B. Consider the situation where the bowling ball is incident on the same system from the left with a
kinetic energy of 40 J, as shown below.
C
B
KE = 40 J
h
A
D
Use the first column on the sheet titled ‘Reference Sheet – Bowling Ball’ to:

Record the potential, kinetic, and total energy of the ball at points A, B, C, and D.

Note how, if at all, the ball’s speed changes as it moves through the system.

Describe where in the system you are most likely to observe the ball (you might use the ‘100
frames of video’ idea here.)

Sketch a picture graph of the probability density of the bowling ball between points A and D.
(Make sure you label your axes!)
C. The experiment is repeated, but this time the incident bowling ball has a kinetic energy of 20 J, as
shown below. Fill in the second column on the reference sheet for this scenario.
C
B
KE = 20 J
h
A
D
Look back at the graphed you sketched in question A. This potential energy arrangement, where
there is a higher potential energy in some area than in its surroundings, is called a potential energy
barrier. (Contrast this with the potential energy well, where the potential energy in some area is
lower than its surroundings.)
Tunneling – a Quantum Model Consequence
p.181
II. Extending the Cart Model for Electrons
For several weeks now, we’ve been building a model of electrons that includes wave-like properties. This
is not by accident – scientists believe the wave-particle model to be the best in describing the actual
behavior of electrons.
However, one hallmark of a good model is that it is simple, and it would certainly
be easier to think about electrons as just particles (really tiny carts, perhaps). In
fact, it wasn’t too long ago that the particle model was the prevalent idea regarding
electrons. However, as we saw in a previous lab, using the cart between two walls
to model an electron in a potential well has limitations – it worked for some ideas,
but not for others.
In this section, we’re going to return to using a cart,
and see if it’s a good model to use in thinking about
electrons interacting with potential energy barriers.
To do so, we’ll use the magnet cart setup that we first
saw in the ‘Energy and Probability’ tutorial.
N
S
N
N
S
S
S
S
N
S
N
S
N
S
N
N
S
N
N
S
S
N
S
N
S
N
S
N
N
S
A. Go to the magnet cart setup in the front of the room.
1. Practice rolling the cart through the magnets until you can start it at a speed that allows it to just
make it though. Use the computer to produce a kinetic energy picture graph of your experiment.
2. Reproduce the kinetic energy picture graph on the axes below. Label your axes. Label the values
of your axes – we haven’t made you do this much previously, but for this experiment, it’s
important.
Return to your table so another group can use the apparatus.
3. Sketch a potential energy picture graph on the axes below. Label your axes. Label the values of
your axes as well.
p.182
Tunneling – a Quantum Model Consequence
4. Sketch a picture graph of probability density for the magnet cart. Label your axes.
B. Consider an cart incident on the potential barrier, but with less energy.
1. What was the kinetic energy of the cart at the beginning of its motion (that is, before it went past
the magnets)?
2. What would happen to the cart if you started it with half as much kinetic energy? (If no other
group is using the apparatus, return and check your predictions.)
3. Sketch a picture graph of probability density for the magnet cart in this scenario. Label your
axes.
How is this graph different from the one you sketched in A.4?
4. Is there any chance, given the initial energy you gave the cart in the second experiment, that the
cart will make it past the magnets? Explain.
Tunneling – a Quantum Model Consequence
p.183
All right, you didn’t think we’d let you go a whole tutorial without wave functions, did you? We’re going
to explore the predictions of quantum physics for potential energy barriers, then return and contrast them
with the predictions of the ‘cart’ model.
III. A Scanning-Tunneling Microscope
The scanning tunneling microscope (STM), invented in 1981, is a device widely used in both industrial
and fundamental research to obtain atomic-scale images of metal surfaces. It provides a three-dimensional
profile of the surface that is very useful for characterizing surface roughness, observing surface defects,
and determining the size and conformation of molecules on the surface.
The Laboratory for Surface Science and Technology
(LASST) at the University of Maine uses a scanningtunneling microscope and other similar technologies to
study the surfaces of materials they develop. You can find
out more about them at http://www.umaine.edu/lasst, or
visit them in the new wing of Barrows Hall.
In this section, we’re going to develop a model of how a
scanning-tunneling microscope works. Very crudely, a
STM consists of a very pointed tip that passes over the
surface of a material, but does not touch it. The closer the
tip is to the material, the more electrons are measured in
the tip. This is a very surprising effect, since previous
physics models predicted that they shouldn’t be there!
M
A
T
E
R
I
A
L
TIP
GAP
Imagine that 100 electrons are measured in some material. The tip of the scanning-tunneling microscope
is brought close to the surface, and 25 electrons are measured in the tip.
A. We’re going to start thinking about this system in terms of probability.
1. In the first box on the ‘Reference Sheet – Scanning Tunneling Microscope’, sketch a probability
density picture graph for this system.
2. The tip is moved farther away from the surface, and now only 10 electrons are measured in the
tip. How does this change your graph of probability density? Sketch the new graph on the
reference sheet.
3. The tip is now moved closer to the surface than it originally was, and now 40 electrons are
measured in the tip. How does this change your graph of probability density? Sketch the new
graph on the reference sheet.
Tunneling – a Quantum Model Consequence
p.184
B. We can refine our model a bit by working with some energy values. A very rough model of this
scanning-tunneling microscope system is given by the values below:
0 eV Material

PEsystem  30 eV
Gap
0 eV
Tip

1. 100 electrons, each with TE = 20 eV, are measured at the surface of some material. 25 of them
are subsequently measured in the tip.

a. How much kinetic energy does each electron have while it is in the material? How do you
know?
b. How much kinetic energy does each electron that is found in the tip have? Explain your
reasoning.
2. Two students (pretend they’re members of your group – you may even want to have two of your
group members read the statements out loud) are discussing this situation, and make the
following statements:
Student 1: “The barrier’s energy is higher than the electron’s
energy, so the electrons that make it through the barrier
lose energy in the process. They’ll probably lose about
half, meaning the electrons that are in the tip will have
about 10 eV of energy.”
Student 2: “No, energy is conserved, so the electrons in the tip
will have the same energy, the same  - everything about
them is the same. That means the same probability density
in the tip – wait, then shouldn’t all the electrons make it
to the tip? I don’t get it.”
Discuss the student statements with the members of your group. With which parts of the
statements do you agree? With which parts do you disagree? Record your comments in the
space below.
Tunneling – a Quantum Model Consequence
p.185
3. What type of wave function would the electrons have in the material region? How do you know?
4. What type of wave function would electrons have in the tip region? How do you know?
a. How would wavelength in the tip region compare to the wavelength in the material region?
b. How would amplitude in the tip region compare to the amplitude in the material region?
5. What type of wave functions must exist in the gap region? Support your answer by considering
and analyzing…
a. … the energy of the electron and energy of the system.
b. … the amplitudes in the material and tip regions.
Keep in mind that your answers must be consistent when answering the same question from
two different perspectives.
6. Sketch the wave functions for each scenario on the reference sheet, making sure to change your
sketches appropriately for each situation.
C. Imagine that the gap was somehow changed – filled with a different gas, perhaps, so that the potential
energy of the system in the gap is 50 eV. Everything else stays the same.
1. How, if at all, would this change  in…
a. …the material?
b. …the gap?
c. …the tip?
Be sure to think about the type of function, the amplitude, the wavelength, and the curviness at a
given value in each region!
Tunneling – a Quantum Model Consequence
p.186
2. Sketch a picture graph of the wave function for this new situation.
(x)
MATERIAL
GAP
TIP
The phenomenon that we’re describing with this wave function graph is called tunneling. As we’ve
previously said, the energy arrangement where the potential energy is higher in some region than its
surroundings is called a potential energy barrier.
These words, tunneling and barrier, with obvious links to our everyday usage, often lead to confusion
about what’s going on. We want you to be careful about what you do and don’t mean when using them.
D. We think of barriers as obstacles that are difficult or impossible to get through.
1. Would classical (“cart”) electrons make it to the tip? How do you know?
2. This barrier does involve loss, but it’s not the loss that many students link it with.
a. What, specifically, is lost?
b. What, specifically, is not lost?
Tunneling – a Quantum Model Consequence
p.187
IV. Board Meeting 1 - Mapping surfaces
We’ve spent some time on the tunneling part – now we’re going to
explore the scanning part of the scanning tunneling microscope
name.
Researchers at IBM were able to manipulate atoms on a surface and
produce the image seen at right, where each of the points you see
represents the location of a single atom. The image of this
arrangement was then produced by information gathered by a
scanning-tunneling microscope.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
The following questions should be addressed by all groups:
Imagine you were using a scanning-tunneling microscope to probe this surface. How could you tell that
the surface spelled out I-B-M? What specifically would you have to do? Be sure to address such issues
as…
1. Moving the tip
2. Moving the surface.
3. The type(s) of information you’d have to collect.
4. How you would interpret that information.
V. Cart-like Electrons vs. Quantum Electrons
Compare the predictions about the behavior of the ‘cart’ electron to those of the quantum electron.
A. In what ways, if any, is the behavior of the ‘cart’ electron similar to the behavior of the quantum
electron?
B. In what ways, if any, is the behavior of the ‘cart’ electron different than the behavior of the quantum
electron?
C. What evidence do we have that we have to choose one description over the other?
Tunneling – a Quantum Model Consequence
p.188
VI. When Quantum Applies – deBroglie Wavelengths
A. Have you ever observed a bowling ball to be where the laws of classical physics predict it cannot be?
That is, if you roll it at a wall, would you ever expect to observe it on the other side of the wall?
Quantum physics seems to apply in some situations, like electrons at relatively small energies
(10’s of eVs), but we never see bowling balls travel where they’re not supposed to go. To explore the
reason, we introduce something physicists call the deBroglie wavelength, which all objects have:
deBroglie  A 
1
KE
where A is a number that depends on the amount of mass something has, and KE is the object’s
kinetic energy in Joules. The table below lists the value of A for some objects:
 Object
A (Jskg-0.5)
Electron
5.0  10-19
Bowling Ball
2.0  10-34
Earth
2.0  10-46
(Don’t worry too much about the units; if you calculate a deBroglie wavelength with this formula, the
units of the wavelength turn out to be meters.)
B. Calculate the deBroglie wavelength for an electron with a kinetic energy of 20 eV, a bowling ball
with a kinetic energy of 20 J, and the Earth, with a kinetic energy of 2.7  1033 J. To do so, follow
these steps:
1. If necessary, convert the energy to Joules (recall that 1 eV = 1.6  10-19 J).
2. Take the square root of the kinetic energy.
3. Find the inverse of the square root of the kinetic energy.
4. Multiply the inverse of the square root of the kinetic energy by the appropriate constant A.
1
Object
Energy (eV)
Electron
20 eV
Energy (J)

Bowling ball
20 J
Earth
2.7  1033 J
KE
KE


 1 
A  

 KE 
Tunneling – a Quantum Model Consequence
p.189
One idea that we can use to discern whether or not we’ll see an object behave in manners predicted by
quantum physics is to examine the size of its deBroglie wavelength. If the deBroglie wavelength is on the
order of a length that can be discerned by some measuring device, we might be able to observe the
quantum properties of the object. If not, we have no way of knowing whether or not the object is
behaving like a quantum object.
For reference, an ordinary microscope cannot resolve objects much smaller than 10-6 meters. An electron
microscope can resolve objects down to about 10-10 meters.
C. Is the deBroglie wavelength for a bowling ball on the order of a length that can be experimentally
discerned? Do we expect to be able to observe the quantum behavior of a bowling ball?
D. What about the deBroglie wavelength for an electron? Is it on the order of a length that can be
experimentally discerned? Do we expect to be able to observe the quantum behavior of a bowling
ball?
E. Given enough time, do you expect that technology will advance to the point where one would be able
to observe the quantum properties of everyday objects (i.e. bowling balls)? Why or why not?
If you’re interested in this topic, there are some researchers who are actually finding quantum phenomena
in “large” objects, and don’t believe this deBroglie wavelength argument. You can read more at
http://www.quantum.univie.ac.at/.
VII. When Quantum Applies – Curviness Arguments
There’s another way we can reason about when to apply quantum ideas. Although we haven’t said so
explicitly, it turns out that an object’s wave function depends on mass. In the Schrödinger Equation
k  TEparticle  PEsystem x Curv ( x)
k depends on mass. If m doubles, so does k.
A. What happens to the curviness of the wave function as mass increases? What about the wavelength?

Tunneling – a Quantum Model Consequence
p.190
B. Earlier in lab, you sketched the wave function for 20 eV electrons that tunnel from the material to the
tip, where 25 are detected for each 100 at the surface.
1. For comparison purposes (to what? Hang on…), reproduce that sketch here.
(x)
MATERIAL
GAP
TIP
2. Imagine (OK, its not very physical!) that you had an identical system and a 20 eV bowling ball.
The bowling ball’s mass, however, is about 1031 times greater than the electron’s. Sketch what
the bowling ball’s wave function would look like on the axes below:
(x)
Region A
Region B
Region C
3. What’s the probability of finding the bowling ball in Region C?
VIII.
Final Board meeting
Check your answers on section VII in a board meeting. Present them on the white boards and discuss
until all agree on the answers that you have found and how you arrived at those answers.