Download Quantum physics in school

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Eger M 1992 Hermeneutics and science education:
an introduction Sci. Educ. 1 (4) 337—48
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(Cambridge: Cambridge University Press)
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and Teaching in Scottish Secondary Schools: The
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qualitative study of attitude toward science Sci.
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Kingdoms (New York: St Martin’s Press)
Quantum physics in school
I Lawrence King’s School, Worcester, UK
A teaching route for introducing quantum ideas
into the school classroom using modern devices is
presented. This has been developed by a team of
academics and school-teachers and trialled once.
The ideas are quantization, wave–particle duality,
nonlocality and tunnelling.
This article attempts to map out a minimum
approach to quantum ideas as they might be
developed for a year 12 group (these students are
age 17, in the first year of a two-year pre-university
course). The trial group was working on module two
of the Nuffield A-level modular course. The
important elements of the approach are that:
●
●
●
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the ideas developed should be of further use
and reinforce important ideas already met in
physics and elsewhere
the style of argument used should be
identifiably physics
the ideas should have cultural worth
●
the skills acquired by the students should be
intrinsically useful.
The work has grown, and is a part of a larger scheme
to make something of quantum ideas starting from
devices. Parts of this work have been presented at
GIREP in Italy and at the Malvern Update in March
1996. I am indebted to Professor John Jefferson
(DRA, Malvern), Mr Mark Ellse, Mrs Marie Arthur
(King’s School, Worcester), Dr Ian Sage (DRA,
Malvern) and Dr Mike Uren (DRA, Malvern), who
have been part of this development team.
The central ideas developed in this article are those
of:
● quantization
● wave–particle duality
● nonlocality
● tunnelling.
Each of these is developed with the help of
plausibility arguments which draw on experiences in
the laboratory, demonstrations, discussions and prior
R E T H I N K I N G P O S T - 16 P H Y S I C S
knowledge. The aim is to make the quantum view of
the world:
● intelligible
● plausible
● fruitful [1]
What follows is a possible route through some
simple ideas for introducing these concepts in a way
which seems to make them assimilable and
challenging. My thanks to my current lower sixth
(L6) who admirably fulfilled the roles of guinea-pigs!
so that the ideas become part of students’ cultural
toolkit and not just a set of standard responses or
specialised tools for particular problem situations.
Period 1
Figure 1. Apparatus for measuring Planck’s
constant using multiple coloured LEDs.
Figure 2. Graph of students’ results from the
experiment to measure Planck’s constant.
The first and central idea to get across is the idea of
quantization. To do this I chose a couple of visible
props familiar to the pupils from their work to GCSE
and from their everyday environment: an
incandescent filament lamp and an LED. With these
two one can develop a model which suggests that
energy only comes in packets.
We first established a link between colour and
potential difference using the incandescent lamp.
(The notion of potential difference is a hard one and
this course was trialled with able students in the
second term of their L6 — still the revision
engendered by this approach was welcomed.
Students need to feel comfortable with an idea to
use it as the basis for modelling in unfamiliar
situations.) This enabled me to move onto the idea
of a potential drop and the energy that is available
for transfer to light. This idea can be reinforced by
looking at the changes in colour of a short length of
resistance wire when the pd across it is varied. An
LED is then introduced. This has a very different
behaviour but the key element to bring out is that
there is a minimum potential drop which will
produce light and that the resultant light is of one
colour only. Some of the reason for this is bound up
in the construction of the LED (this is further
developed in the work which we have been doing on
introducing quantum ideas via devices but did not
trouble the students in trials) but the main thrust of
the argument is to establish the phenomenological
link between the colour and the pd. Discussion
smoothly transfers this to a link between energy and
Table 1. Tabulated results from the experiment to measure Planck’s constant.
Striking pd (V) Energy per electron (aJ) h (10234 J s)
LED colour
Wavelength (nm)
Frequency (PHz)
blue
470
0.638
2.38
0.381
6.0
green
563
0.533
1.69
0.270
5.1
yellow
585
0.512
1.63
0.261
5.1
orange
620
0.483
1.48
0.237
4.9
red
650
0.462
1.47
0.235
5.1
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wavelength. Reinforcement follows by getting the
class to suggest what the pd might be for a green
LED and then trying it out. At this stage there seems
to be some grounds for a link between wavelength
and pd (energy of emitted light) and it is therefore
worth trying these ideas out on a selection of LEDs.
(This brought the first 35 minute lesson to a close.)
Period 2
The next double period was spent looking at a range
of LEDs after a review of the state of play at the
beginning of the period. We used LEDs from 470 nm
to 650 nm, which are readily available and not too
expensive, and I have suggested a way for wiring
them up elsewhere (figure 1) which seems to have
withstood the destructive integral of sixth formers
with respect to time [2].
After a little inspection of the results, plotted using
the spreadsheet on a Psion 3a palm-top computer,
the well heeled decided that really the graph should
be linear, wavelength clearly was not, and so we had
better try frequency. They were much happier with
this!
And so we had to look for a relationship between
frequency and energy per electron and found h,
Planck’s constant. This double period was a
generous time allocation but allowed time for their
expectations to play a part, a result with a small but
messy amount of data, and a clear-cut consensus to
emerge from competing research groups who could
walk round, comparing and then modifying and
remeasuring, until happy with the graphs on their
palm-tops (figure 2 and table 1). We ended with eV
5 hf and the photon hypothesis.
Figure 3. Reverse-biased LEDs with a magic wand.
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At this stage the students are fairly convinced that
light is emitted as photons, but showing them that it
also arrives as photons can be both entertaining and
convincing. The trick is simply to suggest reversing
the LED. If reverse polarized, then electrons will not
pass through because they have a potential barrier.
The hill the electrons dropped down to release the
photons now blocks their path in the reverse
direction. But this hill can only be surmounted by
supplying the electrons with a particular amount of
energy — a quantum of energy — equal to the energy
that was emitted when a photon was emitted. Such
a supply of readily quantized energy happens to be
available — a pocket torch. This can then be tried out
by directing light from the incandescent filament
lamp of the torch though a standard red filter onto
red and green LEDs with the expected results (to the
teacher not to the students — they should believe
before the event that success in this matter is highly
unlikely!). The process can then be repeated with a
standard green filter and with red and green LEDs.
Period 3
The third period starts with explanation and
exploration of the photon hypothesis as to how we
might explicate changing intensities as well as
different colours, and indeed why we do not see
light as grainy. Not the 'size' of the photons but a
calculation of the number emitted by a 40 W lamp
every second is a convincing argument against
graininess.
This idea of quantization can be embedded further
by using an ultra bright LED wired as a magic wand
and going along the row of different LEDs used for
Figure 4. Using the Quantex Infrared Sensor Card.
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Figure 5. A guide to tasks to link to a more conventional treatment of quantization.
measuring Planck’s constant when they are reversebiased (figure 3).
Picking out only the matching frequency nicely
illustrates the concept of resonance. Further
exploration involves the use of an IR sensor which
does not apparently produce any output but which,
when shone on a Quantex Infrared Sensor Card [3],
produces a discernible glow (figure 4). This leads to
a nice discussion of visible and invisible photons and
where the energy might come from in order to
produce the glow. The card must actually be
prepared for use before detecting IR light: 'charge
this card with regular ambient (or fluorescent) light
prior to use)' are the instructions printed on the
reverse.
Figure 6. A facsimile of the BBC micro screen.
Period 4
As a link to more conventional treatments (even the
one on the syllabus!) they were then given a series
of tasks to wrap up this first section.
This BBC Micro simulation (figure 6) is a minimal
approach to manipulating variables so as to
investigate relationships! When the pd (V to set),
Figure 7. The potential drop across an LED.
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wavelength (L to set), metal (M to set) are selected
and R is pressed the electron leaves the metal
when the photon is incident and heads off towards
the cathode for appropriate values of L and M.
You then adjust V until the electron only just makes
it across the gap, and this gives you the kinetic
energy of the electron.
behave. This is a metal–insulator–metal (MIM)
junction which is used in some displays. If we join a
metal to an insulator then the electrons are free to
move within the metal but cannot move into the
insulator. This a very similar situation to electrons
not moving from right to left in the LED above. We
can represent this as a potential barrier (figure 8).
This sequence of activities established quantization
firmly: light has both wavelike and particle-like
properties. If what are held to be waves have
particle like properties, might it not be the case that
particles also have wavelike properties? The next
phenomenon we looked at was tunnelling, which
gave evidence for this idea as well as introducing
the two important ideas of nonlocality and the
superposition of states.
To look at the nature of electrons we will look at a
thin insulator sandwich, the metal–insulator–metal
junction. The easiest way to do this in the school
laboratory is with a capacitor (figure 9).
Period 5
The first period started with recapitulation. Coming
from the work on LEDs students have the idea that
changes in the electron’s energy levels can be
related to a potential jump (figure 7).
For electrons travelling from the right to the left the
step is a potential barrier: energy must be supplied
to the electrons in the form of photons for the
electrons to surmount the barrier. To explore the
behaviour of electrons we need a readily controllable
supply of electrons — the best source is a metal
wire! We chose an arrangement that helped us to
develop our current understanding of how electrons
Figure 8. The energy map for a metal-insulator
junction.
So we try it out: The result is quite surprising — as we
increase the pd we find a small current, which then
rises nonlinearly instead of what we might have
expected: no current at all then suddenly plenty of
current. There is a stage where some of the electrons
get though and some do not. This is in spite of the
fact that the simple expectation is that all the
electrons are at the same energy level. To
understand this we must first look at what is
happening to the energy map that we drew above as
the pd is increased. The pd drop can only happen
across the capacitor — wires have no resistance —
and so as we increase the pd the map changes as
shown in figure 10.
Notice that we are not altering the structure of the
metal or the insulator and therefore the electrons are
incident on the barrier at the same height and the
barriers between metal and insulator are also the
same height. The effect of increasing the pd is
Figure 9. The potential map and circuit for a MIM
junction.
Figure 10. How the energy map changes as the pd increases.
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therefore to make the barrier appear thinner to the
electrons. As we make the barrier thinner an
increasing number of electrons seem to leak through.
Classical physics, i.e. the physics of electrons as small
particles, has carried us a long way but has no
answer to this. In the same way that when we were
stuck with light as a wave we looked at the particle
model, so now we could look at the wave model for
help with particles.
Figure 11. Apparatus for looking at exponential
decay of microwaves through a barrier.
At this stage it is a nice illustration of having the
right idea at the right time to introduce the (then)
relatively unknown, Prince Louis de Broglie, who
produced this very novel idea in 1924 as part of his
PhD thesis: that electrons should display wavelike
behaviour. The suggested relationship for the
wavelength of particles is
Table 2. Table of results from exponential decay of
microwaves through a barrier.
Books
Reading (mA)
Ratio
15
0.9
0.82
14
1.1
0.79
13
1.4
0.82
12
1.7
0.81
11
2.1
0.78
10
2.7
0.71
9
3.8
0.88
8
4.3
0.80
7
5.4
0.86
6
6.3
0.82
5
7.7
0.86
4
9.0
standard deviation
0.05
l 5 h/mv.
Anything with a momentum that is very much larger
than h will not display wavelike effects that are
readily observable. We are then left with the
problem of illustrating what it might be like for an
electron to have wavelike properties. The things that
we know about electrons in a wire at this stage (low
Figure 13. Locating an electron.
Figure 12. Results of exponential decay of microwaves through a barrier.
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Figure 14. Simulating the interaction of an electron with a potential barrier.
velocity, small mass) make it likely that electrons will
have a small momentum, which will of course help
us. This is where the MIM barrier helps. This was the
end of the first period.
Period 6
The next double period needs to resolve the puzzles
at the end of the last. A good place to start is with
propagation of waves through a barrier to see if they
behave in a way that will help us to understand the
leaking of electrons through a potential barrier.
Microwaves seemed to be a suitable source of waves
and we found a use for school physics texts by
utilizing them as a variable thickness barrier (figure
11).
The decay of intensity (remember that this is
proportional to the square of the amplitude) with
the thickness of the barrier is exponential. The
results obtained by my students are shown in figure
12 and table 2.
To pull things together we introduced the idea of
locating an electron. We have never been able to
pinpoint an electron — it was only the earlier
arrogance of physicists that suggested electrons
might be a point. We therefore can position an
electron more truthfully by saying where it is likely to
be (figure 13).
We drew a probability wave to show where it is likely
to be. The evolution of this probability wave over
time then determines where the electron is likely to
be found. To find out how an electron is likely to
behave when it collides with a barrier we need to
know what happens to a wave that is incident on a
barrier. We had some experimental knowledge of
this already and compared this with a simulation of
the interaction of a probability wave describing the
Figure 15. The interaction of an electron with a wider potential barrier.
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position of an electron with a barrier [4]. I have
extracted the frames shown in figure 14. The
exponential decay as the wave is incident on the
barrier was spotted by the students (honest!).
Uncertainty in position, the electron 'being in two
places at once', particle likeness, wave likeness,
incomplete picture models were all buzzing around.
Time for a break!
This is a model of a single collision of an electron
with a MIM barrier. Note that there is a definite
probability that the electron goes through and a
definite probability that it does not. This is not as a
result of our lack of knowledge about the system —
as far as we can tell this is really what happens. The
electron is said to be in a superposition of states: it
is on both the left and the right hand sides of the
barrier. Interpreting this state of affairs for a single
electron is hard work and led to discussions of
paradoxes such as Schrödinger’s Cat, but for an
assembly of electrons fired at the barrier the problem
is easier — only a fraction will get through. If we
make the barrier thicker (figure 15) then more of the
waveform has decayed by the time the other side is
reached and therefore fewer electrons penetrate.
This resulted in a fairly heated discussion — there
was some feeling that the student had been led up
the garden path but they began to see a coherent
picture for the quantum world emerging.
Period 7
Figure 16. Simulating the interaction of an alpha
particle with a fixed nucleus.
Figure 17. A nuclear potential.
Again to complete the picture, for the penultimate
period we built in historical links for this work.
Starting with de Broglie, whose opinion was
regarded as a little extreme by the examining
committee which had to refer to Einstein for a
second opinion, we then showed the diffraction of
electrons [5]. He was shown to be right in 1927 by
streams of electrons showing diffraction patterns;
these could only be convincingly explained by wave
ideas. It was in fact these very patterns that had led
to the rejection of the particle model of light in the
19th century.
Period 8
We then allowed some time for exploration and
development in the final period of the two weeks.
Development 1. The reason why ordinary particles
such as humans, cars, buildings, cricket balls do not
display wavelike properties is linked to Planck’s
constant h. A lively discussion followed Gamow in
allowing Planck’s constant to grow. Tunnelling is a
nice basis on which to discuss this phenomenon and
allows plenty of room for lively engagement and
(fairly!) disciplined use of the imagination.
Development 2. Tunnelling and alpha emission.
Another process that has a fixed probability of
occurring but the actual occurrence of which is
unpredictable is the emission of particles from
radioactive nuclei. Here again we cannot say with
certainty what the outcome will be for any one
alpha particle in any one atom but we can say with
a great deal of certainty what the outcome will be
when aggregated over a large number of particles.
An additional problem which points to tunnelling is
that the emitted alpha particles are of
approximately 5 MeV whereas the bombardment of
atoms with alpha particles shows that the energy
needed to penetrate a nucleus is greater than 9
MeV. Geiger and Marsden’s experiment — about
which a reminder was issued by using a simulation
(home-grown simulation utilizing Interactive Physics
[6], as advertised in Physics Education!) — provided
the evidence for this when Rutherford was
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postulating his nuclear atom (see figure 16).
Conclusion
The idea of quantum tunnelling allowed this
paradox to be resolved. The alpha particle can
tunnel through the barrier shown in figure 17. Now
that we know more about the nucleus we can
calculate the height of the potential barrier. The
classical equivalent of this is any roller coaster. If it
has enough energy to climb a hill then it should be
emitted with a kinetic energy equal to the potential
energy at the top of the hill.
In this account I have sought to give not only an
outline of a route but also a feeling for the
achievability of the route in a UK school with live(ly)
year 12 students. The opportunities are there in this
material, not only to develop quantum ideas without
too much historical detritus, but also to utilize the
material in a way that fits into the current syllabus
structure. Many students might well want to take
these ideas further. There are two more layers of the
onion under preparation — a more rigorous
development of the treatment of semiconductor
devices, and a treatment of the band theory of
solids. These impart all the depth and rigour one
could hope for and are part of the complete
development which was alluded to at the beginning.
In year 13 it may well be appropriate to reinforce
some of this work by looking at:
● the wave nature of electrons again using
electron diffraction
● interference of single photons
● electron waves in atoms
● more complex devices such as laser and
resonant tunnelling diodes, which incorporate
plenty of physics.
The first three of these are well covered by the
Nuffield A-level Physics course [7]; the last one has
been the concern of the development group.
Another way of looking at this problem is to regard
it as an extension of the uncertainty principle: you
can borrow a bit of energy for a while — the more
energy the shorter the while. In this the case the
trade-off is between the trip time to get through the
barrier and the energy that is needed to climb over
the barrier. This is often quoted as DEDt 5 h.
Note that a simple qualitative prediction from this
model is that the higher energy alpha particles
should come from the more radioactive nuclei — this
is the case.
Development 3. The scanning tunnelling electron
microscope. Scanning tunnelling microscopy is a
technique developed in the 1980s and allows the
imaging of solid surfaces with high resolution. The
operation of a scanning tunnelling microscope
(STM) is based on a tunnelling current, which starts
to flow when a sharp tip approaches a conducting
surface at a distance of approximately one
nanometre. This current varies very sensitively with
the insulating gap through which it tunnels and
therefore we must control the tip position in such a
way that the tunnelling current and, hence, the
tip–surface distance are kept constant The tip is
mounted on a piezoelectric crystal which expands a
small amount when a voltage is applied at its
electrodes. The tip scans a small area of the sample
surface whilst maintaining this constant distance.
This three-dimensional movement is recorded and
can be displayed as an image of the surface. Under
ideal circumstances, the individual atoms of a
surface can be resolved and displayed.
Any of these last three can be used as a starter for a
brief research report. The first requires a disciplined
imagination and the second two can link strongly
with other areas of the curriculum.
286
Received 24 May 1996
References
[1] Osborne R and Freyberg P 1985 Learning in
Science (Auckland: Heinemann) p 47
[2] School Science Review December 1993 75 (271)
p111
[3] Available from Quantex, 2, Research Court,
Rockville, MD 20850, USA
[4] Quantum Scattering from Physics Academic
Software, NCSU, Box 8202, Raleigh, NC 276958202, USA
[5] Revised Nuffield A-Level: Teachers Guide 2 1986
(Harlow: Longman) experiment L7
[6] Interactive Physics from Knowledge Revolution,
66 Bover Road, Suite 200, San Mateo, CA 94402,
USA
[7] Revised Nuffield A-Level: Teachers Guide 2 1986
(Harlow: Longman)