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Uncertainty in the
classroom—teaching quantum
physics
K E Johansson and D Milstead
Fysikum, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
Abstract
The teaching of the Heisenberg uncertainty principle provides one of those
rare moments when science appears to contradict everyday life experiences,
sparking the curiosity of the interested student. Written at a level appropriate
for an able high school student, this article provides ideas for introducing the
uncertainty principle and showing how it can be used to elucidate many
topics in modern physics.
Introduction
Physics syllabuses for pupils in the final few
years of their high school studies, e.g. A-levels
in the UK system, include the physics of waves
and provide an introduction to many topics in
quantum physics. The students are thus equipped
to understand at a basic level one of the most
thought-provoking results in modern physics: the
Heisenberg uncertainty principle.
One form
of the Heisenberg uncertainty principle asserts
that, contrary to our expectations from classical
physics, it is impossible to simultaneously know
the position and momentum of a particle to
arbitrary accuracy, whilst another form relates
uncertainties in energy with time. The Heisenberg
uncertainty principle therefore has profound
scientific and philosophical implications.
There already exist a number of studies
discussing the appropriate way to teach quantum
mechanics topics at high school (see, for
example, [1–4]). As a complement to these works,
we provide here a short, self-contained article
specifically on the uncertainty principle. We show
how it can be demonstrated and how it can be used
to gain a deeper understanding of various physical
phenomena such as radioactivity and exchange
forces. It is hoped that this article could be of
interest both for physics teachers and high school
students and could be a resource for schemes such
as the UK’s Gifted and Talented program.
Getting to the uncertainty principle
There are a number of ways to arrive at the
Heisenberg uncertainty principle. One approach
is to use a gedanken (thought) experiment method
to demonstrate that even with ‘perfect’ apparatus
it is impossible in principle to simultaneously
determine a particle’s position and momentum
to arbitrary accuracy [5]. Other authors adopt
a more rigorous approach, demonstrating how
fundamental uncertainties arise from dispersions
in a particle’s wavefunction [6]. Always mindful
that the longer the explanation, the fewer attentive
students, we feel that it is best to start with a
situation with which the students are already very
familiar. Therefore, we use single-slit diffraction
to demonstrate the effects of the Heisenberg
uncertainty principle and show that it is nothing
more than wave–particle duality cast in a different
light. Furthermore, with this approach (outlined
below) there is little new mathematics to consider.
0031-9120/08/020173+07$30.00 © 2008 IOP Publishing Ltd
PHYSICS EDUCATION
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K E Johansson and D Milstead
_
e
_
e
_
e Δx
py
_
e
_
e
B
θ
px
A
not to scale
Figure 1. Single slit diffraction with an electron beam.
A broad beam of electrons with momentum
p y moves in the y -direction towards a wall
containing a small slit of width l , as shown in
figure 1. Far behind the slit is a detector which
can detect a single electron. We know that any
electrons which are detected must have passed
through the slit. When we look at the locations
of the scattered electrons in the electron detector,
we observe that they are spread out in a diffraction
pattern with maxima and minima.
Of course, we are only witnessing the
consequences of wave–particle duality, something
with which most students are already familiar. An
electron has a de Broglie wavelength λ = phy
related to the momentum p y it possessed as it
entered the slit. As long as the wavelength is
of the appropriate size, we obtain an observable
single-slit diffraction pattern. The experiment can
be rerun with photons, protons, pions, or indeed
any sub-atomic particle and similar results are
obtained.
However, as a complement to the wave picture, the diffraction experiment can be understood
as a consequence of the fundamental limit nature
imposes on how well we can know the position of
any particle x and its momentum p x .1 When
the electron is within the slit, x is simply the slit
width. It is impossible to say exactly where it is
but it must be somewhere between the slit walls.
The Heisenberg uncertainty principle states that if
nature allows us to gain knowledge about the position of a particle we must lose information on
its momentum, a consequence which is contrary
1
Heisenberg uncertainty principle relations can also be
written for the position and momentum in the y and z directions
although, for simplicity, only uncertainty in one dimension is
considered throughout this article.
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PHYSICS EDUCATION
to our expectations from classical physics. However, therefore, there must be a spread of values
of momentum in the x -direction p x , explaining
why the electron paths are spread. We are allowed
to know roughly by how much the momentum can
change and we can see that some values (and hence
electron paths) are more probable than others. For
example, an electron is more likely to be found
in the region around point A than point B in figure 1. However, we can never know in advance
which value of momentum an individual electron
will take. Thus, nature will never allow us to simultaneously know the position and momentum of
a particle to arbitrary precision. This is a handwaving mathematics-free way of introducing the
Heisenberg uncertainty principle.
To write the uncertainty principle mathematically can be easily done using our electron experiment. We can see that the momentum spread occurs in the x -direction and is mostly restricted such
that the electrons tend to fall near to the central
peak (the maximum of the distribution). The study
of diffraction tells us that the scattering angle of
λ
the first minimum can be approximated as θ ≈ x
,
where λ is the de Broglie wavelength of the incoming electron, given by λ = phy ; here p y is the
incoming electron’s momentum. Taking the angle
between the maximum and the first minimum as
λ
the angular uncertainty, we find that θ ≈ x
.
The angle θ is related to the momentum of the
x
electron as it leaves the slit: θ ≈ p
, where
py
px is the magnitude of the scattered electron’s
momentum in the x -direction, which is required if
it were to come close to the first minimum. Thereλ
x
fore, we find that p
≈ x
= p yhx . The unpy
certainty in momentum and position can thus be
written xp x ≈ h . This confirms our earlier
assertion that nature will not let us simultaneously
know the exact values of a particle’s position and
momentum.
To achieve high accuracy in the position we
need to narrow the slit, but at a price of a greater
uncertainty in the momentum. The Heisenberg
uncertainty principle cannot, of course, be used to
predict all the features of the diffraction pattern,
for example, the locations and intensities of all
of the maxima and minima. However, it can
explain the fundamental observation that particles
will deviate from their path, and it will also give
an approximate estimate of the location of the first
minimum.
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Uncertainty in the classroom—teaching quantum physics
A formal derivation of the Heisenberg
uncertainty principle using the full machinery of
quantum mechanics gives a more refined and
general formula than that which we obtained for
single-slit diffraction
xpx h
.
4π
Applications of the Heisenberg uncertainty
principle
The Heisenberg uncertainty principle is valid for
any particle in any situation, not only for a
specific experiment such as single-slit diffraction
of electrons. If we have any knowledge of the
uncertainty in a position then the Heisenberg
uncertainty principle provides information on the
uncertainty in momentum and vice versa. For
example, an electron bound within a hydrogen
atom is known to be confined across a distance
that is about the size of the atomic radius. The
Heisenberg uncertainty principle allows us to
estimate the minimum momentum of such an
electron.
The Heisenberg uncertainty principle can
also elucidate the motion of particles at low
temperature. It is popularly and wrongly supposed
that at 0 K all motion stops and that even the
electrons in atoms are somehow stationary. The
Heisenberg uncertainty principle can easily show
this statement to be fallacious. As the above
example showed, if a particle is confined within
a finite distance x then it must have a minimum
spread in momentum p 4πhx , which cannot
be zero. Hence, there must be some motion even
at 0 K. The lowest possible energy of a particle is
known as the zero-point energy, and according to
quantum mechanics this cannot be zero.
Does a particle possess a definite position
and momentum?
It is tempting to suppose that the Heisenberg
uncertainty principle must be wrong, and that it
must be possible to simultaneously know exactly
the position and momentum of a particle. For
80 years physicists have attempted to design an
experiment which would breach the Heisenberg
uncertainty principle. Many ingenious gedanken
experiments were dreamed up by the early
quantum pioneers, including Einstein, Heisenberg,
March 2008
and Bohr [7, 8].
Even assuming ‘perfect’
apparatus they found it was impossible to design
an experiment which would measure the values of
the position and momentum of a particle exactly.
Since we can neither predict nor measure
them, is it even worth worrying about whether or
not a particle has definite values of its position
and momentum? It was Heisenberg’s view that
physics should only concern itself with measurable
quantities. Einstein disagreed and thought that
the failure of the Heisenberg uncertainty principle
and quantum mechanics to predict definite
values of position and momentum shows that
quantum mechanics is somehow an incomplete
theory. However, in the most common view of
quantum mechanics, the so-called Copenhagen
interpretation, a particle simply does not possess
specific values of position and momentum, but
the act of measurement forces it to take a
value. This issue remains an open and hotly
debated question in quantum mechanics. Niels
Bohr preferred to call the Heisenberg uncertainty
principle the indeterminacy principle, to stress that
it is really impossible to determine the quantities
to an arbitrary precision, independent of the
experimental set-up.
Another uncertainty principle: energy and
time
An experiment at a particle collider measures the
mass of a Z particle. The text books report
that the Z mass is measured to several decimal
places: 91.1876 GeV/c2 . Hence, if the experiment
measures the mass of a single Z boson and finds
it to be 88 GeV/c2 , do we assume that there is
something wrong with the detector? Actually,
no. If we measure the masses of a large number
of Z particles we see a broad distribution which
peaks at 91.1876 GeV/c2 (as shown in figure 2)
but which has a large population even at 88 and
94 GeV/c2 .2 This spread of mass values is due
to the very short lifetime of the Z particle (of the
order 10−25 s). A particle with such a short lifetime
simply does not have a single, well-defined mass;
as for position and momentum, there is a spread of
possible values M one can obtain when making
a measurement. We cannot predict in advance with
which mass value an individual Z particle will be
2
The Hands on CERN education program [9] contains a large
number of particle collisions from the LEP electron–positron
collider, which can be used to further explore the Z0 particle.
PHYSICS EDUCATION
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K E Johansson and D Milstead
A
No. Z bosons
A
B
C
B
ΔM
time
A
B
Figure 3. An interaction taking place via a particle
exchange.
88
92
90
Z mass (GeV/c2)
94
Figure 2. Measurement of the Z-mass made at the LEP
electron–positron collider.
produced but, as figure 2 shows, some values are
more likely than others.
Einstein’s equation relates the energy of a
particle to its mass via E = mc2 . We can use the
Z mass experiment to demonstrate another form
of the Heisenberg uncertainty principle: Et ≈
h
. Here, E = mc2 is effectively the
4π
uncertainty in the mass (around 2 GeV/c2 for the Z
particle) and t is the particle’s mean lifetime
and not an uncertainty in time, which is anyway
a meaningless concept in non-relativistic physics.
Quantum mechanics allows us to write a
general expression for the energy–time uncertainty
principle: Et 4hπ , which relates the
minimum uncertainty in energy E of any
particle system undergoing a change during a time
interval t . This naı̈vely implies that nature can
allow extremely large energy changes as long as
E 4πht . In practice, as we saw with the
measurements of the Z, nature prefers to keep
energy shifts to a minimum (Et ≈ 4hπ ). We
therefore use the minimum uncertainty relation
to discuss (below) radioactivity and differences
between the fundamental forces.
Understanding fundamental forces with
the Heisenberg uncertainty principle
The energy–time relation can also help us
understand forces between particles.
The
traditional picture, included in high school
syllabuses, is that particles exchange another
particle in order to transmit a force, as shown
in figure 3, where particles A and B undergo
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repulsion via the exchange of particle C. Such
exchanged particles are examples of virtual
particles, so called since they are never directly
observed and only exist for an unmeasurably short
time. However, our naı̈ve picture of particle
exchange has several problems. One difficulty is
that if particle A is at rest and emits the exchange
particle then energy conservation can be violated:
the mass of A < mass of B + mass of exchange
particle C.3 This leads many authors to assert that
nature allows energy E to be ‘borrowed’ (the
mass of the exchanged particle) for a short time t
whilst the interaction takes place. However, since
we have already abandoned much of our familiar
classical physics as we enter the sub-atomic world
3 Another flaw is that our picture can only be easily used
to visualize repulsive forces: particle A recoils after emitting
the exchange particle which moves to kick particle B further
away. Attractive forces can, however, be accommodated
by considering the exchange particle to be emitted from A
in a direction away from B, thereby pushing A towards B.
Furthermore, owing to the uncertainty in the position of the
exchange particle (the Heisenberg uncertainty principle again),
it may well ‘reappear’ to the right of B and push it towards A.
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Uncertainty in the classroom—teaching quantum physics
it is comforting to know that conservation of
energy still holds. As we saw with the Z decay,
particle states which exist for a short time are
allowed to take various values of mass. The shortlived virtual particles can possess a range of mass
values as long as energy is always conserved. We
shall use the ‘borrowing of energy’ technique here
when discussing the forces since it is simple and
convenient. However, it should be remembered
that it is nothing more than a shorthand way of
hiding the mathematics involved with calculating
the masses of the various virtual particles.
The strong nuclear force is traditionally
viewed as the exchange of pions between nucleons
(protons and neutrons). A proton emits a pion
which is later absorbed by a neutron. The energy
‘borrowed’ must be the pion rest mass energy
E = m π c2 = 0.14 GeV, and this regulates
how far the pion can travel: ct ≈ 4πhcE =
force4 , which keeps the nucleus together. A typical
energy of an α -particle is 4 MeV, which is far
less than the energy required to overcome that
force field and escape. Therefore, α -decay should
not happen. However, we observe α -decay, and
people die because of cancer caused by α -particles
ejected from nuclei. How does this happen?
The Heisenberg uncertainty principle allows an
α -particle to occasionally ‘tunnel’ through the
energy barrier and into the region in which only the
electric repulsion force is relevant. Brushing under
the carpet the complicated interactions of the α particle with the force field, it is possible to view
the Heisenberg uncertainty principle as allowing
the α -particle to ‘borrow’ sufficient energy E for
the short time t needed to escape. The more
energy needed to be borrowed, E ≈ 4πht , the
shorter the time available for borrowing and the
less likely it is to escape.
6.6×10−34 ×(3×108 )
4×3.14×0.14×109×1.6×10−19
= 7 × 10−16 ≈ 10−15 m,
which is consistent with inner nuclear distances. If
nature had chosen the pion mass to be smaller we
could be dealing with nuclei of far greater sizes!
We can use similar reasoning to understand
the range of the weak and electromagnetic forces.
As mentioned earlier, the Z boson has a mass
of 91 GeV, implying that it can be exchanged
over tiny distances of ct ≈ 4πhcE =
6.6×10−34 ×(3×108)
4×3.14×91×109×1.602×10−19
≈ 10−18 m.
This shows the very short range of the weak
force, which also explains why weak processes are
so unlikely at low energies, where the ‘borrowed’
energy is so large.
The electromagnetic force, unlike the weak
force, is mediated by photons which are massless,
and the range of this force is hence infinite.
Barrier penetration
In our classical world, a ball rolling along a surface
is unable to overcome a barrier (see figure 4) if its
kinetic energy at the foot of the barrier 12 mv 2 is
less than the gravitational potential energy mgh it
would possess at the top. This can also be regarded
as the ball not possessing sufficient energy to pass
through a gravitational force field. An analogous
situation happens in a nucleus, but this time there
is a different ending to the story. Approximately
1022 times per second, an α -particle, a bound state
of two protons and two neutrons, bounces off the
walls of a force field created by the strong nuclear
March 2008
Consequences of the Heisenberg
uncertainty principle
A deterministic view of the world holds that if
only we knew all of the laws of physics and
could measure positions and properties of each
particle then we could ultimately predict how the
universe, and presumably human beings, would
behave with time (assuming we had the necessary
computing power!). The uncertainty principle
conflicts with this, since it forbids us to know the
exact values of the position and momentum of any
particle. If we cannot know the present, then we
cannot predict the future! Quantum mechanics
gives us uncertainties and probabilities for various
outcomes and nothing more. Thus, if quantum
mechanics is the correct theory for the sub-atomic
domain then we must accept that our model of
nature is not a like wound clock but instead more
like a game of dice.
Although the game of dice is played out at
small distances it could affect our macroscopic
world as well. For example, as discussed earlier,
the α -decay of a nucleus is permitted by the
uncertainty principle, but we cannot predict when
this will happen for any individual nucleus. Should
4
If it is not obvious why the α -particle should have any motion
at all then recall the discussion on zero-point energy. An α particle is known to be localized within a finite distance (the
size of a nucleus) and therefore must have non-zero momentum
according to the position–momentum form of the Heisenberg
uncertainty principle.
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K E Johansson and D Milstead
Table 1. Summary of physical phenomena which can be understood/calculated using the Heisenberg uncertainty
principle.
Physical phenomenon
Single-slit diffraction
Zero-point energy
Exchange particles
α -decay
Explanation with the Heisenberg
uncertainty principle
xp 4hπ , a particle position is
determined to an accuracy of the slit-width
x = l , implying a spread in momentum
xp 4hπ , x = 0 leading to non-zero
energy
Et 4hπ , short-lived particles can
possess a range of mass values. Can also be
viewed as nature ‘borrowing energy’ E
for a time t to allow exchange processes
to happen which would otherwise violate
energy conservation. This also explains the
range of various forces
Et 4hπ an α -particle borrows energy
for a short time to escape a force field
a person be unfortunate enough to be in the vicinity
of a radioactive source when it emits a burst of
α -particles, then this may cause a fatal cancer to
develop.
The Heisenberg uncertainty principle in
the classroom
In the classroom the scattering of electrons not
only demonstrates the wave nature of the electron,
that is the wave–particle duality related to quantum
mechanics, but also the Heisenberg uncertainty
principle relating to position and momentum at
work. Diffraction of light, easily accessible at
school, can also be interpreted as a consequence
of the Heisenberg uncertainty principle. The mass
width of particles, particularly striking for the
W and Z particles [9], is a manifestation of the
Heisenberg uncertainty principle relating energy
and time. The effect is present also in nuclear
and atomic physics, but as the lifetime of the
particle states are normally much longer, the effect
is smaller. In addition, the Heisenberg uncertainty
principle can help in understanding nuclear decays
and barrier penetration and also particle decays
and interactions as particle exchange mechanism.
However, for a detailed description of particle
interactions, the dynamics of the standard model
of particles is needed.
Discussion
Many authors and popularizers of science assign
different meanings to the terms x, p x , E, t .
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Many texts prefer to introduce the position–
momentum form of the Heisenberg uncertainty
principle using a gedanken approach in which x
is clearly an experimental resolution and p x a
momentum ‘kick’ received by a particle. Others
prefer more rigorous derivations in which x and
px represent the dispersion of the particle wavefunction. We feel that the most suitable approach
is simply to refer to x and p x as the limits
nature will impose on our knowledge of position
and momentum. We used the paradigm of the already familiar single-slit diffraction and stressed
that there is no physical situation, for example an
experiment or a bound atomic electron, for which
the Heisenberg uncertainty principle is not valid. It
is not necessary to introduce the concept of a particle wavefunction and the consequent discussion
of superposition of states. Such an approach does
not so easily provide the final expression of the
Heisenberg uncertainty principle and is best left to
a rigorous undergraduate course.
However, it is important to be aware of the
limitations of using the Heisenberg uncertainty
principle blindly. As observed in one of the more
erudite texts on particle physics, if a physicist
uses the uncertainty principle to put forward an
argument the listener should ‘keep his hands on his
wallet’ [10]. The same argument which was used
to explain the infinite range of the electromagnetic
force fails when applied, for example, to explain
why exchanges of gluons only take place within
hadrons, as the role of the strong charge, the
colour charge, also affects the interaction range.
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Uncertainty in the classroom—teaching quantum physics
Such occasional inconsistencies are to be expected
owing to the impossibility of simplifying quantum
field theory to such a basic level.
Summary
The Heisenberg uncertainty principle (or indeterminacy principle) is an important and thoughtprovoking part of quantum mechanics. It is also
a very powerful tool; two short inequalities can be
used to elucidate a number of physical phenomena,
as summarized in table 1. Used with care, it enables approximate calculations to be made of otherwise intractable situations. Several of the consequences of the Heisenberg uncertainty principle
can be demonstrated in the classroom, and it is also
an excellent starting point for a discussion on the
bizarre nature of the quantum world.
Received 4 September 2007, in final form 24 October 2007
doi:10.1088/0031-9120/43/2/006
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March 2008
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Erik Johansson is a Professor at
Stockholm University and Director
emeritus of Stockholm House of Science.
He is the co-coordinator of the Education
and Outreach project of the ATLAS
experiment at CERN. He shared the first
European particle physics outreach award
in 2001, and received a Webby award in
the science category in 2005 for the
Hands on CERN particle physics
education project.
D Milstead is a lecturer at Stockholm
University Physics Department and is, by
trade, a particle physicist. He has done
research at the HERA electron–proton
collider and currently works at the
ATLAS experiment at the LHC at CERN.
In 2003 he won an international Science
Communication prize given by the
Association of Friends and Sponsors of
the Deutsches Elektronen SYnchrotron
Labor (DESY). This was for a television
documentary (on BBC2) about a new
type of particle, known as a magnetic
monopole.
PHYSICS EDUCATION
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