In my previous video, I introduced Heisenberg’s
Uncertainty Principle; I performed a little
home experiment, covered some theory and also
used a few examples to show you how the Uncertainty
Principle works in real life. However, I am
sure many of you were left wondering… But
where does the uncertainty principle come
from? How does it arise? In this video I’ll
discuss in some detail the way in which the
uncertainty principle and wave-particle duality
are intimately related and how uncertainty
naturally arises out of the indeterminacy
which is inherent in wave-like systems. It’s
a fascinating topic – I hope you enjoy it.
We know that both light and matter exhibit
what is known as wave-particle duality. Under
different circumstances, both light and matter
can display either type of behaviour, wave-like
or particle-like, depending on how we set
up our experiment, the types of questions
we decide to ask Nature and whether or not
we make an observation. Let’s take an electron
for instance. How can we possibly describe
such an entity? We know that it acts like
a particle whenever we look at it; however,
when we don’t look, it acts like a wave.
And since an unmeasured electron acts differently
than a measured one, it appears we simply
cannot describe it without referring to the
act of measurement.
Which brings me to the question of whether
it is meaningful at all to ask if something
like an electron does exist in and of itself,
as a particle or as a wave. Is an electron
a wave? Is it a particle? Is it both? Is it
either one or the other but never both at
the same time? Could it be that the reason
these questions are so difficult to answer
is that it may be meaningless to talk about
the notion of reality itself or about the
independent existence of objects in any sort
of absolute sense? Perhaps all we can talk
about is our own perceptual interfaces, our
own conceptual constructs and our own mathematical
models of the world. As Werner Heisenberg
himself very wisely said:
“What we observe is not Nature itself, but
Nature exposed to our method of questioning."
And of course this inevitably brings me to
physicist Niels Bohr, Heisenberg’s fatherly
friend and mentor. Bohr developed a philosophical
view which he termed “complementarity”;
I’ll cover this topic in detail in my next
video but for now it suffices to say that
what Bohr pointed out was the fact that the
properties of a quantum system seem to be
completely dependent on what the observer
chooses to measure; in this way, the questions
we ask and our choice of experimental setup
determine what sorts of properties the system
will manifest and what sort of features it
will exhibit, and this certainly includes
its wave-like and particle-like behaviour.
So, yes, wave-particle duality is a type of
complementarity: how much wave-like or particle-like
we observe a system to behave depends entirely
on our mode of questioning. What we ask, how
we ask it and whether we make a measurement
or not does matter. In fact, Bohr actually
denied that it is even meaningful to talk
about the nature of a system “per se”,
that is, to talk about its properties, in
themselves, independently of them being observed.
Here we are talking not about a reality in
itself, but about a deeply contextualistic
reality, a reality which manifests differently,
in complementary ways, depending on the questions
we ask, a reality which is inseparable from
our instruments of observation and our modes
of questioning.
And yet, despite the possible meaninglessness
in asking some of these questions… or perhaps
I should say thanks to the fact that Niels
Bohr and Werner Heisenberg found it meaningful
to discuss these profound ontological and
epistemological questions, they came up with
their Complementary and Uncertainty Principles.
And it is because of these principles – amongst
many other important contributions – that
Bohr and Heisenberg became two of the main
founders of an incredibly precise mathematical
theory which – believe it or not - successfully
combines the particle-like and the wave-like
features we observe in the world. Two complementary
descriptions of reality beautifully merged
into one single theory. Yes, I am talking
about quantum mechanics, the most precisely
tested theory in the history of science: a
mathematical model which predicts the way
the world manifests as we make observations.
In other words, a theory not predictive of
events but predictive of observations, a theory
which incorporates the concepts of energy
quantisation, complementarity, wave-particle
duality, the uncertainty principle and the
correspondence principle.
Ok, so let’s start this section by talking
about waves. Ask yourself this question: where
is a wave? Where in space-time is a wave located
exactly? Imagine we have a completely uniform
wave stretching out to infinity in all directions.
What is the wave’s exact position? Is it
here or is it there? Is it everywhere? Or…
Is it nowhere? It is obvious that in this
situation the question of “where” - that
is, asking about the wave’s exact position
- makes no sense at all. On the other hand,
it is clear that it still makes perfect sense
to ask how fast and in which direction the
wave is moving; in other words, to ask at
which velocity the wave is propagating.
Now, let’s imagine that the type of wave
we are dealing with is a probability wave.
Not a physical tangible wave that can be perceived
or measured, but a wave that represents a
world of possibility, a wave that describes
how likely we are to observe any given particular
value of a property of the system under study
- such as its position for instance - should
we decide to make a measurement. To illustrate
how this works – by using this type of wave
– we make the rule that the probability
of measuring the electron in a certain position
– let’s say, here - is proportional to
the square of the wave’s amplitude at that
position. So, applying this rule in this particular
example, we can see that the probability of
finding the electron over here is higher than
the probability of finding the electron over
there, since the amplitude of the wave is
larger here than it is over there.
Ok, so it turns out that the wave I described
earlier on – remember, the completely uniform
wave stretching out to infinity – that wave
is actually the simplest kind of probability
wave, and it can be used to describe a particle
which has no forces acting on it; yes, we
are talking about a free particle. How is
that so, you may wonder? Well, when no forces
at all act on a particle, the particle’s
velocity - and therefore its momentum - is
always constant, very well-defined at all
times. The essence of wave-particle duality
is the recognition that all particles of matter
or light also exhibit wave-like behaviour,
and so any given momentum can always be directly
associated with a particular wavelength. In
this case, applying de-Broglie’s formula
to the free electron, we can see that a constant
momentum implies a constant wavelength. And
what does a constant wavelength mean? It means
we have a uniform wave that goes on indefinitely.
So, we now have a wave that can be successfully
used to describe our free electron. Through
the wave’s wavelength we can know the particle’s
momentum. However, here’s the problem. What
about the electron’s position then? If this
kind of wave describes a free particle with
a very well-defined momentum, where is the
particle located then? It’s easy to see
now that a uniform wave with such well-known
wavelength doesn’t really help us describe
a particle that is more likely to be located
in one place than another. In other words,
as soon as we have absolute knowledge of the
particle’s momentum, asking “where”
the particle may be found ceases to be a meaningful
question. It’s just meaningless: there ceases
to be a “where” in this case. Asking about
the particle’s position makes no sense because
the wave is spread out over all space. We
can see therefore that this wave doesn’t
actually make a good model if we want to describe
a particle that is more likely to be found
in a particular location than another. It
is clear that what we need is a varying amplitude,
amongst other changes. So what can we do then?
Well, what we need to do now is find a different
wave-based description of the particle which
also allows us to specify (at least to some
degree of accuracy) the position we are likely
to find the particle should we decide to make
a measurement, at any given time.
The answer to this problem is something called
a travelling wave-packet. And what does a
travelling wave-packet look like? Well, in
one dimension – let’s take the x direction
for instance – we are talking about a short
“burst” of waves that moves along the
x axis. The amplitude of this burst of waves
is large only in the region of the wave packet
and it falls off rapidly on either side. Since
the probability of finding the particle at
a particular location is proportional to the
square of the amplitude, the particle is in
this case likely to be found only in the region
of the wave packet. This solves our problem.
There is still a degree of fuzziness as to
the question of “where” the particle may
be found, but at least we have made some progress.
Ok, so how do we build a wave packet then?
Well, a wave packet can be built up by adding
together a series of infinitely long wave-trains
whose wavelengths lie within a finite range
of values. The mathematical ideas that underlie
this technique were developed by French mathematician
Jean-Baptiste Fourier and the technique is
known as Fourier Synthesis. In simple terms,
a wave packet can be constructed from an infinite
set of sinusoidal waves of different wavelengths,
with phases and amplitudes such that they
interfere constructively only over a small
region of space and destructively elsewhere.
Adding a continuous distribution of waves
of different wavelengths together produces
an interference pattern which begins to localise
the wave; we’ve created a wave-packet.
The wave packet model allows us to take into
account the wave-like properties of the particle
but the important point to notice here is
that – because the model also allows us
to give a more or less localised description
within space-time – it now also helps us
to take into account its particle-like features,
such as position in this case. We have therefore
constructed a model which has at its heart
the wave-particle duality we wanted to describe.
Perfect!
Now, quantum mechanics describes the state
of a system – whether this system is an
electron, a photon or something else – using
probability functions similar to the wave
packets we have just described. And it is
precisely from the very principles underlying
this probabilistic description that aims to
reconcile the particle-like and wave-like
properties in Nature that Heisenberg’s Uncertainty
Principle comes from. Here’s how:
As we have just seen, if we want to describe
the conceptual idea of a free particle with
a well-known momentum, then we have to use
a wave that is infinitely long and uniform
everywhere, which in turn makes it impossible
to state where the particle might be found
at any given time. The concept of “where”
becomes meaningless in this case. So although
we know the particle’s momentum with absolute
precision, we find that we can no longer even
ask where the particle may be found.
We have also seen that it is possible to describe
the particle using a wave packet. In this
case, the particle is fairly localised in
space, so its position can be more or less
defined, but since the wave packet’s building
blocks are a number of waves each of them
having a different wavelength, this results
in the momentum of the particle now being
quite poorly defined. In other words, since
each wave we have used as a building block
for the wave packet has a different wavelength,
the momentum’s uncertainty is now quite
large. It seems that the more we manage to
localise the wave-packet in space, the more
uncertainty we find in the particle’s momentum,
and vice-versa.
A rigorous mathematical derivation of these
facts results in Heisenberg’s Uncertainty
Principle, where the uncertainty in position,
∆x, is approximately the width of the wave
packet, and ∆p is the uncertainty in momentum:
The Uncertainty Principle requires that a
narrow spread in one of these quantities is
offset by a wide spread in the other. There’s
always a trade off in how well-defined these
two variables can be at any given time. For
instance, if ∆x is small, it follows that
∆p is large – at least as large as ℏ
/ (2∆x).
So… now you know how Heisenberg’s Uncertainty
Principle arises! Contrary to popular belief,
the Uncertainty Principle is not the result
of measurement disturbances but it lies at
the heart of quantum mechanics, because quantum
mechanics is the way we mathematically model
the indeterminacy, the wave-particle duality
and the quantisation we encounter in Nature.
The Uncertainty Principle is a result which
can be derived theoretically from the fundamental
principles of quantum mechanics, principles
which describe an inherent indeterminacy and
a trade-off in the relationship between certain
pairs of variables or properties. In physics,
these pairs of properties are called complementary
or conjugate observables. Which brings me
back to the concept of complementarity.
In my next video, I’ll cover Bohr’s Principle
of Complementarity in much more detail, we’ll
talk about complementary observables and how
all of these ideas relate to the Uncertainty
Principle and its derivation. I’ll also
show you other forms of the Uncertainty Principle
which involve other pairs of observables,
such as energy and duration.
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