Today I’ll take a less known historical
view and explore how cloud chambers and the
measurements that take place inside them gave
birth to Heisenberg’s Uncertainty Principle.
We’ll see how the idea that there is such
thing as a particle trajectory is completely
unjustified from the point of view of quantum
mechanics and how the measurement problem
arising in an environment such as a cloud
chamber leads us to the modern concept of
quantum decoherence. Let’s get started!
Heisenberg’s Uncertainty Principle tells
us that we cannot simultaneously define both
the position and the momentum of the particle.
If that’s the case, does it imply that the
notion of a particle trajectory no longer
makes sense? After all, a trajectory is a
clearly defined path where – at the very
least – the position of the particle is
known at every moment in time. If the Uncertainty
Principle holds, does that mean that trajectories
are illusions then?
Quantum mechanics is one of the most successful
theories we have. Evidence of the underlying
probabilistic nature of the Universe - as
described by quantum mechanics - has been
obtained experimentally, again and again over
the years, without failure. As we have seen
in previous videos, the state of a particle
is described by a probabilistic wave in quantum
mechanics. But what is the ontological nature
of a particle, you may wonder? Well, we can
start by asking ourselves: What is a particle,
when some of its properties are not well-defined
at all? Or...Does a particle still exist when
the concepts of a well-defined position or
a well-defined momentum no longer apply? The
more we learn about the quantum world, the
more evident it is that what exists, what
is real is a very elusive concept indeed.
As Neils Bohr very wisely put it: “Everything
we call real is made of things that cannot
be regarded as real”.
So… The idea that the world is made of particles
with well-defined properties or the idea that
these so-called particles follow well-defined
trajectories, are ideas which are simply not
compatible with the fundamental principles
of quantum mechanics, nor with experimental
results. Quantum mechanics does not deal with
well-defined objects, but with wave-like mathematical
entities. What we instinctively think of as
a particle with a definite position or momentum
at any given time, and hence what we think
of as a trajectory, are just emergent constructs,
as we move from the mathematical world of
a given quantum system, to the tangible world
of the so-called classical world, through
the process of making observations.
It seems that our ideas of there being such
a thing as a well-defined particle or such
a thing as a well-defined trajectory come
only as a result of our interpolations, through
the process of measurement, as we attempt
to somehow map the fuzzy quantum micro-world
onto the macro-world we experience, that is,
as we clumsily attempt to reconcile the quantum
mechanical description of reality with information
pertaining to the tangible world of experience.
In this process, somehow, we end up confusing
the world of possibility with the world of
actuality; we conflate the two worlds together
in order to produce a seamless story that
is somehow meaningful to us. In other words,
we mix up the two worlds because we feel the
need to create a consistent story in space-time
that “makes sense” to us. We prefer to
think that – if we measure particle-like
events over here at the beginning of the experiment
and then we also measure particle-like events
over there at the end of the experiment – it
then must be the case that the so-called particle
has actually travelled from here to there
during that time interval, it must be the
case that trajectories are real.
Well, no matter what we’d like to think,
turns out there is actually no justification
whatsoever for thinking that way! According
to quantum mechanics - and certainly this
is how Heisenberg thought about it - the world
of possibility should not be confused with
the world of actuality. And when it comes
to particles, we just cannot think of them
as objects which follow well-defined trajectories.
As Heisenberg argued, there is no way of establishing
what happens between two consecutive measurements,
and therefore, our idea that a real particle
followed a path between those two measurements,
is simply not compatible with quantum mechanics.
In his own words: ‘It is of course tempting
to say that the electron must have been somewhere
between the two observations and that therefore
the electron must have described some kind
of path or orbit even if it may be impossible
to know which path.” Despite any temptation,
Heisenberg maintained that the classical notion
of a particle trajectory being a continuous,
unbroken path, through space-time is completely
unjustified.
What? No trajectories? But hold on a second
– you might object - there are instances
where we can actually observe a particle’s
path! How can you say that there is no such
thing as a well-defined particle trajectory
when we do in fact observe them? An alpha-ray
trace observed in a cloud chamber, for instance,
does indeed look like a trajectory in space-time!
Aren’t these cloud chamber tracks plain
evidence that the uncertainty principle is
actually violated??
Well, first of all - as Heisenberg pointed
out – let’s ask ourselves this question:
Is this track in the cloud chamber really
evidence of a continuous particle trajectory?
Or is it actually nothing more than a series
of ill-defined discrete positions corresponding
to our observation of a series of macroscopic
water droplets simply lined up after each
other? It seems we may get the appearance
or illusion of there being a clearly defined
trajectory when in fact all we have is evidence
of the excitation of a certain finite number
of gas molecules due to a few transfers of
energy by the quantum entity. And we should
not let this mislead us into thinking that
the particle would have pursued a well-defined
space-time trajectory in the absence of the
cloud chamber absorbing molecules.
Secondly – and this is quite important – even
though it appears that the trace in a cloud
chamber is a well-defined trajectory, it turns
out that the water droplets actually have
diameters many orders of magnitude larger
than the electron itself, and therefore the
uncertainty principle isn’t violated at
all, due to the thickness of the path defined
by the water droplets, which is of the order
of microns. In addition, the momenta in cloud
chambers are given by energies of at least
a few keV. Multiplying these two quantities
- after converting to the appropriate units
- results in a quantity which is much larger
than ħ. The Uncertainty Principle always
holds.
The reason why this is a fascinating subject
is that – believe it or not – it was precisely
the discussion of this topic that eventually
led Bohr and Heisenberg to formulate their
Complementarity and Uncertainty Principles.
So-called trajectories within cloud chambers
represent one of the key problems that the
fathers of quantum mechanics – including
Bohr, Heisenberg, Pauli and Schrodinger - discussed
at length when addressing the implications
of their theory. Although they were convinced
that they were right, these physicists realised
how difficult it would be to convince other
leading physicists that they would need to
abandon all attempts to construct spatio-temporal
perceptual models of processes pertaining
to particles. As a matter of fact, the physical
interpretation of quantum mechanics was the
central theme of discussion between Bohr and
Heisenberg.
The key problem of how to reconcile the foundations
of quantum mechanics with the observation
of what looks like the continuous trajectory
of an electron in a cloud chamber kept these
two physicists occupied for weeks on end.
In the 1920s, the cloud chamber was the nearest
anyone could come to seeing an individual
electron, and the visible tracks produced
in such chambers really look like the effects
of fast moving particles. How can we reconcile
this with the foundations of quantum mechanics?
– they thought. Turns out that, thinking
about such cloud chambers was what led Heisenberg
to the formulation of his Uncertainty Principle.
In Heisenberg’s own words:
“We had always said so glibly that the path
of the electron in the cloud chamber could
be observed. But perhaps what we really observed
was something much less. Perhaps we merely
saw a series of discrete and ill-defined spots
through which the electron seemed to have
passed. In fact, all we do see in the cloud
chamber are individual water droplets which
must certainly be much larger than the electron.
[…] A brief calculation after my return
to the Institute showed that one could indeed
represent such situations mathematically,
and that the approximations are governed by
what would later be called the uncertainty
principle of quantum mechanics”.
Another important thing to consider is that,
a macroscopically observable – seemingly
classical - trajectory of a particle can also
be conceptualised as the result of repeated
collapses of the wave function due to repeated
measurements, in this case, within the context
of the cloud chamber. Classical observables
such as position and momentum – and hence
what we call a trajectory - can only be properties
emerging as a consequence of the interaction
between the microscopic system and a macroscopic
measurement apparatus and the observer who
uses it.
As we have seen, by no means are position
or momentum to be considered properties already
possessed by the system before measurement.
Hence, the argument goes that - if we observe
something which appears to be a particle trajectory
- it must consist of a series of discrete
measurements performed quickly in succession
one after the other, whereby the momentum
or position observables have been brought
from the realm of potentiality to the realm
of actuality, always within the limits of
Heisenberg’s Uncertainty Principle.
But what constitutes an observation? - you
may wonder. And who or what is performing
these series of measurements in the cloud
chamber then? Is the presence of liquid droplets
sufficient? Is a measurement apparatus all
we need or do we need a conscious observer?
And what constitutes the system and what constitutes
the environment? Is it arbitrary? Well, we
have indeed come to the very heart of what
physicists call the Measurement Problem in
quantum mechanics. Today, it remains unsolved.
These are incredibly important questions.
The kind of questions which Bohr, Heisenberg
and the rest of the founding fathers of quantum
theory spent hours pondering about. Even today,
nearly 100 years later, there is still no
consensus as to how one should go about solving
these problems.
I just find it fascinating how the issue of
attempting to describe what happens inside
a cloud chamber highlights the subtlest discrepancies
in our interpretation of quantum mechanics,
bringing to the surface not only Heisenberg’s
Uncertainty Principle but, equally importantly,
the most relevant questions which inevitably
lead us to the very heart of the Measurement
Problem.
If you’d like to dig deeper into this fascinating
topic and explore how quantum mechanics predicts
the appearance of cloud chamber traces without
referring in any way to the classical view
that the particle has at any time a well-defined
position and therefore without assuming that
it travels along a definite trajectory, just
google “Mott Analysis of Cloud Chamber Problem”.
For your reference, I have put a link below
in the description area to a very interesting
paper which touches on all these ideas, including
the history behind it.
Here, I’ll give you a short – if perhaps
a bit unsatisfactory – summary regarding
the ideas behind this explanation as it concerns
cloud chambers and trajectories:
In its very essence, all that quantum mechanics
does is give predictive rules informing us
of what is likely to be observed. This is
important: we do not have information relating
to what happens, but relating to what is likely
to be observed should we make a measurement.
Inevitably, the wording of these rules involves
words such as ‘particle’ or ‘wave’
which – let’s not forget – strongly
reflect our limitations as we attempt to objectify
the world we experience as well as give some
sort of meaning to all of that which we do
not experience, namely, the world of possibility
reflected in our quantum mechanical equations.
We should therefore not get too attached to
our vocabulary, or at least do our best not
to give it any absolute sense.
Right to, so back the problem at hand. Let’s
say we are studying some radiation - say we
are dealing with a cosmic ray - which interacts
with an atomic object, such as water droplets
in a cloud chamber. In quantum mechanics,
all we can talk about is the probability that
a particular water droplet will appear as
being excited. And of course this goes linked
to the probability that bubbles will appear
in the vicinity of this object. Well, turns
out this probability is extremely small when
any two atoms are not aligned in the direction
along which the radiation is propagated, whereas
the probability is quite appreciable when
they are. This idea is easily generalised
to three, four, or any number of atoms. And
this results in an overwhelming majority of
cases where the excited water atoms will lie
along a particular direction, let’s say
vertical, and therefore a vertical trace will
be observed. In a nutshell, this is how quantum
mechanics avoids the concept of trajectory
all together when it comes to predicting what
will be observed inside a cloud chamber.
Quantum mechanics correctly accounts for the
observed phenomena, but remarkably, it does
so without referring in any way to the classical
view that the incident cosmic particle has
at any time a well-defined position and thus
without the assumption that it travels along
a definite trajectory. Hence, there is no
justification whatsoever in concluding that
our experiment provides evidence that, before
the particle interacted with atoms within
the chamber, it had a definite trajectory
in space-time. In fact, we can’t even say
it had a definite trajectory within the chamber,
despite the observed trace! Heisenberg’s
Uncertainty Principle always holds. We simply
cannot talk about the existence of a definite
particle with definite properties nor about
such particle having a trajectory; even less
make conjectures about what happened within
space-time before our measurements took place!
I’d like to finish this video by mentioning
that it is precisely here, within the context
of attempting to describe what takes place
inside a cloud chamber - that the modern idea
of decoherence has its early roots. Indeed,
decoherence in quantum mechanics can be used
to understand what we observe inside a cloud
chamber.
But what is decoherence? Decoherence results
from applying the basic quantum mechanical
axioms to the fact that macroscopic systems
always appreciably interact with their environment,
including their “internal” one. Decoherence
provides an explanation for the way a quantum
system loses information into the environment,
and thus it shows how classical physics emerges
from quantum physics.
The problem of explaining classical behaviour
in quantum systems – such as the appearance
of a classical trajectory within a cloud chamber
– was rediscovered around the 1980s, when
remarkable progress was made in the exploration
of the classical/quantum border. This led
to the development of decoherence theory,
which is based on the idea that we need to
construct theoretical models of a system plus
its environment in order to study, quantify
and analise the emergence of classical behaviour
in a quantum system.
Decoherence effects prove to have a major
role concerning our apprehension of the world;
in particular, in accounting for the fact
that macroscopic objects are always seen as
localised (that is, situated in definite places).
Quantum decoherence is not a new theory, but
simply provides a satisfactory explanation
for the observation of classical behaviour,
as the quantum nature of the system leaks
into the environment.
So… summarising, quantum mechanics deals
with wave-functions, mathematical entities
which are not physical objects at all, but
probability functions that describe a world
of possibility, and this fuzzy world of possibility
doesn’t involve neither particles nor well-defined
trajectories. Not only it is unnecessary to
insist that a so-called particle must follow
a classical trajectory in space-time to get
from one point to another but in quantum mechanics
it has NO MEANING to make such a claim. In
quantum mechanics, we can only speak about
the probabilities of observing a particle’s
interactions at certain locations, hence we
can only speak about the probabilities of
the particle appearing to take one path or
another, should we choose to make a measurement.
The hard core of quantum theory is the set
of its observational predictions; it is important
to remember that by no means it is a theory
that describes what takes place.
As physicist Bernard d’Espagnat explained,
the thought may again and again come to our
mind: But how can that be? No trajectories!
But look here, look at the evidence! “What
about the observed traces? Isn’t the claim
that trajectories do not exist blatantly at
odds with the fact that, undoubtedly, we see
traces? No, it is not.”
So… that’s the way it is, whether we like
it or not. You might say: “I don't believe
it! It's too crazy! I'm not going to accept
it.” I’ll leave you with Richard Feynman’s
words… one thing I have no doubt about is
that Nature surely has better imagination
than we do.
Thank you for watching and thank you so much
for your support. See you very soon!
