Quantum Physics: Chapter 2: Classical Physics

Classical Physics

Before we look at what quantum mechanics has to say about how we are to understand the natural
world, it is useful to have a look at what the classical physics perspective is on this.

According to classical physics, by which we mean pre-quantum physics, it is essentially taken for granted that there is an ‘objectively real world’ out there, one whose properties, and whose very existence, is totally indifferent to whether or not we exist. These ideas of classical physics are not tied to any one person – it appears to be the world-view of Galileo, Newton, Laplace, Einstein and many other scientists and thinkers – and in all likelihood reflects an intuitive understanding of reality, at least in the Western world.

This view of classical physics can be referred to as ‘objective reality’. Observed path Calculated path
Figure 1.1: Comparison of observed and calculated paths of a tennis ball according to classical physics Within this view of reality, we can speak about a particle moving through space, such as a tennis ball flying through the air, as if it has, at any time, a definite position and velocity. Moreover, it
would have that definite position and velocity whether or not there was anyone or anything monitoring its behaviour. After all, these are properties of the tennis ball, not something attributable
to our measurement efforts.

Well, that is the classical way of looking at things. It is then up to us to decide whether or not we want to measure this pre-existing position and velocity. They both have definite values at any instant in time, but it is totally a function of our experimental ingenuity whether or not we can measure these values, and the level of precision to which we can measure them. There is an implicit belief that by refining our experiments — e.g. by measuring to to the 100th decimal place, then the 1000th, then the 10000th — we are getting closer and closer to the values of the position and velocity that the particle ‘really’ has. There is no law of physics, at least according to classical physics, that says that we definitely cannot determine these values to as many decimal places as we desire – the only limitation is, once again, our experimental ingenuity.

We can also, in principle, calculate, with unlimited accuracy, the future behaviour of any
physical system by solving Newton’s equations, Maxwell’s equations and so on. In practice, there
are limits to accuracy of measurement and/or calculation, but in principle there are no such limits.
1.1.1 Classical Randomness and Ignorance of Information Of course, we recognise, for a macroscopic object, that we cannot hope to measure all the positions and velocities of all the particles making such an object. In the instance of a litre of air in a bottle at room temperature, there are something like 1026 particles whizzing around in the bottle, colliding with one another and with the walls of the bottle. There is no way of ever being able to measure the position and velocities of each one of these gas particles at some instant in time. But that does not stop us from believing that each particle does in fact possess a definite position and velocity at each instant. It is just too difficult to get at the information.

In the end, we accept a certain level of ignorance about the possible information that we could, in
principle, have about the gas. Because of this, we cannot hope to make accurate predictions about
what the future behaviour of the gas is going to be. We compensate for this ignorance by using
statistical methods to work out the chances of the gas particles behaving in various possible ways.
For instance, it is possible to show that the chances of all the gas particles spontaneously rushing
to one end of the bottle is something like 1 in 101026 – appallingly unlikely.


The use of statistical methods to deal with a situation involving ignorance of complete information
is reminiscent of what a punter betting on a horse race has to do. In the absence of complete
information about each of the horses in the race, the state of mind of the jockeys, the state of the
track, what the weather is going to do in the next half hour and any of a myriad other possible influences on the outcome of the race, the best that any punter can do is assign odds on each horse
winning according to what information is at hand, and bet accordingly. If, on the other hand, the
punter knew everything beforehand, the outcome of the race is totally foreordained in the mind of
the punter, so (s)he could make a bet that was guaranteed to win.


According to classical physics, the situation is the same when it comes to, for instance, the evolution
of the whole universe. If we knew at some instant all the positions and all the velocities of all
the particles making up the universe, and all the forces that can act between these particles, then
we ought to be able to calculate the entire future history of the universe. Even if we cannot carry
out such a calculation, the sheer fact that, in principle, it could be done, tells us that the future of
the universe is already ordained. This prospect was first proposed by the mathematical physicist
Pierre-Simon Laplace (1749-1827) and is hence known as Laplacian determinism, and in some
sense represents the classical view of the world taken to its most extreme limits. So there is no
such thing, in classical physics, as true randomness. Any uncertainty we experience is purely a
consequence of our ignorance – things only appear random because we do not have enough information to make precise predictions. Nevertheless, behind the scenes, everything is evolving in an entirely preordained way – everything is deterministic, there is no such thing as making a decision,
free will is merely an illusion!!!

The classical world-view works fine at the everyday (macroscopic) level – much of modern engineering relies on this – but there are things at the macroscopic level that cannot be understood
using classical physics, these including the colour of a heated object, the existence of solid objects

. . . . So where does classical physics come unstuck?

Non-classical behaviour is most readily observed for microscopic systems – atoms and molecules,
but is in fact present at all scales. The sort of behaviour exhibited by microscopic systems that are
indicators of a failure of classical physics are
• Intrinsic Randomness
• Interference phenomena (e.g. particles acting like waves)
• Entanglement
Intrinsic Randomness It is impossible to prepare any physical system in such a way that all
its physical attributes are precisely specified at the same time – e.g. we cannot pin down both
the position and the momentum of a particle at the same time. If we trap a particle in a tiny
box, thereby giving us a precise idea of its position, and then measure its velocity, we find, after
many repetitions of the experiment, that the velocity of the particle always varies in a random
fashion from one measurement to the next. For instance, for an electron trapped in a box 1 micron
in size, the velocity of the electron can be measured to vary by at least ±50 ms−1 . Refinement of the experiment cannot result in this randomness being reduced — it can never be removed,
and making the box even tinier just makes the situation worse. More generally, it is found that
for any experiment repeated under exactly identical conditions there will always be some physical
quantity, some physical property of the systems making up the experiment, which, when measured,
will always yield randomly varying results from one run of the experiment to the next. This is
not because we do a lousy job when setting up the experiment or carrying out the measurement.
The randomness is irreducible: it cannot be totally removed by improvement in experimental
technique.

What this is essentially telling us is that nature places limits on how much information we can
gather about any physical system. We apparently cannot know with precision as much about
a system as we thought we could according to classical physics. This tempts us to ask if this
missing information is still there, but merely inaccessible to us for some reason. For instance,
does a particle whose position is known also have a precise momentum (or velocity), but we
simply cannot measure its value? It appears that in fact this information is not missing – it is
not there in the first place. Thus the randomness that is seen to occur is not a reflection of our
ignorance of some information. It is not randomness that can be resolved and made deterministic
by digging deeper to get at missing information – it is apparently ‘uncaused’ random behaviour.
Interference Microscopic physical systems can behave as if they are doing mutually exclusive
things at the same time. The best known example of this is the famous two slit experiment in which
electrons are fired, one at a time, at a screen in which there are two narrow slits. The electrons are
observed to strike an observation screen placed beyond the screen with the slits. What is expected
is that the electrons will strike this second screen in regions immediately opposite the two slits.
What is observed is that the electrons arriving at this observation screen tend to arrive in preferred
locations that are found to have all the characteristics of a wave-like interference pattern, i.e. the
pattern formed as would be observed if it were waves (e.g. light waves) being directed towards the
slits.

The detailed nature of the interference pattern is determined by the separation of the slits: increasing
this separation produces a finer interference pattern. This seems to suggest that an electron,
which, being a particle, can only go through one slit or the other, somehow has ‘knowledge’ of the
position of the other slit. If it did not have that information, then it is hard to see how the electron
could arrive on the observation screen in such a manner as to produce a pattern whose features
are directly determined by the slit separation! And yet, if the slit through which each electron
passes is observed in some fashion, the interference pattern disappears – the electrons strike the
screen at positions directly opposite the slits! The uncomfortable conclusion that is forced on us
is that if the path of the electron is not observed then, in some sense, it passes through both slits
much as waves do, and ultimately falls on the observation screen in such a way as to produce an
interference pattern, once again, much as waves do.

This propensity for quantum system to behave as if they can be two places at once, or more
generally in different states at the same time, is termed ‘the superposition of states’ and is a singular
property of quantum systems that leads to the formulation of a mathematical description based on
the ideas of vector spaces.

Entanglement Suppose for reasons known only to yourself that while sitting in a hotel room
in Sydney looking at a pair of shoes that you really regret buying, you decided to send one of the
pair to a friend in Brisbane, and the other to a friend in Melbourne, without observing which shoe
went where. It would not come as a surprise to hear that if the friend in Melbourne discovered
that the shoe they received was a left shoe, then the shoe that made it to Brisbane was a right shoe,
and vice versa. If this strange habit of splitting up perfectly good pairs of shoes and sending one
at random to Brisbane and the other to Melbourne were repeated many times, then while it is not
possible to predict for sure what the friend in, say Brisbane, will observe on receipt of a shoe, it
is nevertheless always the case that the results observed in Brisbane and Melbourne were always
perfectly correlated – a left shoe paired off with a right shoe.

Similar experiments can be undertaken with atomic particles, though it is the spins of pairs of
particles that are paired off: each is spinning in exactly the opposite fashion to the other, so that the
total angular momentum is zero. Measurements are then made of the spin of each particle when it
arrives in Brisbane, or in Melbourne. Here it is not so simple as measuring whether or not the spins
are equal and opposite, i.e. it goes beyond the simple example of left or right shoe, but the idea
is nevertheless to measure the correlations between the spins of the particles. As was shown by
John Bell, it is possible for the spinning particles to be prepared in states for which the correlation
between these measured spin values is greater than what classical physics permits. The systems
are in an ‘entangled state’, a quantum state that has no classical analogue. This is a conclusion
that is experimentally testable via Bell’s inequalities, and has been overwhelmingly confirmed.
Amongst other things it seems to suggest the two systems are ‘communicating’ instantaneously,
i.e. faster than the speed of light which is inconsistent with Einstein’s theory of relativity. As it
turns out, it can be shown that there is no faster-than-light communication at play here. But it can
be argued that this result forces us to the conclusion that physical systems acquire some (maybe
all?) properties only through the act of observation, e.g. a particle does not ‘really’ have a specific
position until it is measured.

The sorts of quantum mechanical behaviour seen in the three instances discussed above are believed
to be common to all physical systems. So what is quantum mechanics? It is saying something
about all physical systems. Quantum mechanics is not a physical theory specific to a limited
range of physical systems i.e. it is not a theory that applies only to atoms and molecules and the
like. It is a meta-theory. At its heart, quantum mechanics is a set of fundamental principles that
constrain the form of physical theories themselves, whether it be a theory describing the mechanical
properties of matter as given by Newton’s laws of motion, or describing the properties of
the electromagnetic field, as contained in Maxwell’s equations or any other conceivable theory.
Another example of a meta-theory is relativity — both special and general — which places strict
conditions on the properties of space and time. In other words, space and time must be treated in
all (fundamental) physical theories in a way that is consistent with the edicts of relativity.
To what aspect of all physical theories do the principles of quantum mechanics apply? The principles
must apply to theories as diverse as Newton’s Laws describing the mechanical properties of
matter, Maxwell’s equations describing the electromagnetic field, the laws of thermodynamics –
what is the common feature? The answer lies in noting how a theory in physics is formulated.

1.3 Observation, Information and the Theories of Physics

Modern physical theories are not arrived at by pure thought (except, maybe, general relativity).
The common feature of all physical theories is that they deal with the information that we can obtain
about physical systems through experiment, or observation. For instance, Maxwell’s equations
for the electromagnetic field are little more than a succinct summary of the observed properties of
electric and magnetic fields and any associated charges and currents. These equations were abstracted
from the results of innumerable experiments performed over centuries, along with some
clever interpolation on the part of Maxwell. Similar comments could be made about Newton’s
laws of motion, or thermodynamics. Data is collected, either by casual observation or controlled
experiment on, for instance the motion of physical objects, or on the temperature, pressure, volume
of solids, liquids, or gases and so on. Within this data, regularities are observed which are

best summarized as equations:
F = ma — Newton’s second law;
∇ × E = −
∂B
∂t
— One of Maxwell’s equations (Faraday’s law);
PV = NkT — Ideal gas law (not really a fundamental law)
What these equations represent are relationships between information gained by observation of
various physical systems and as such are a succinct way of summarizing the relationship between
the data, or the information, collected about a physical system. The laws are expressed in a manner
consistent with how we understand the world from the view point of classical physics in that the
symbols replace precisely known or knowable values of the physical quantities they represent.
There is no uncertainty or randomness as a consequence of our ignorance of information about a
system implicit in any of these equations. Moreover, classical physics says that this information is
a faithful representation of what is ‘really’ going on in the physical world. These might be called
the ‘classical laws of information’ implicit in classical physics.

What these pre-quantum experimenters were not to know was that the information they were
gathering was not refined enough to show that there were fundamental limitations to the accuracy
with which they could measure physical properties. Moreover, there was some information that
they might have taken for granted as being accessible, simply by trying hard enough, but which we
now know could not have been obtained at all! There was in operation unsuspected laws of nature
that placed constraints on the information that could be obtained about any physical system. In
the absence in the data of any evidence of these laws of nature, the information that was gathered
was ultimately organised into mathematical statements that constituted classical laws of physics:
Maxwell’s equations, or Newton’s laws of motion. But in the late nineteenth century and on
into the twentieth century, experimental evidence began to accrue that suggested that there was
something seriously amiss with the classical laws of physics: the data could no longer be fitted to
the equations, or, in other words, the theory could not explain the observed experimental results.
The choice was clear: either modify the existing theories, or formulate new ones. It was the latter
approach that succeeded. Ultimately, what was formulated was a new set of laws of nature, the
laws of quantum mechanics, which were essentially a set of laws concerning the information that
could be gained about the physical world.

These are not the same laws as implicit in classical physics. For instance, there are limits on
the information that can be gained about a physical system. For instance, if in an experiment we
measure the position x of a particle with an accuracy1 of ∆x, and then measure the momentum p
of the particle we find that the result for p randomly varies from one run of the experiment to the
next, spread over a range ∆p. But there is still law here. Quantum mechanics tells us that
∆x∆p ≥
1
2
~ — the Heisenberg Uncertainty Relation
Quantum mechanics also tells us how this information is processed e.g. as a system evolves in
time (the Schrodinger equation) or what results might be obtained in in a randomly varying way ¨
in a measurement. Quantum mechanics is a theory of information, quantum information theory.
What are the consequences? First, it seems that we lose the apparent certainty and determinism
of classical physics, this being replaced by uncertainty and randomness. This randomness is not
due to our inadequacies as experimenters — it is built into the very fabric of the physical world.
But on the positive side, these quantum laws mean that physical systems can do so much more
within these restrictions. A particle with position or momentum uncertain by amounts ∆x and ∆p
means we do not quite know where it is, or how fast it is going, and we can never know this. But
1Accuracy indicates closeness to the true value, precision is the repeatability or reproducibility of the measurement.

the particle can be doing a lot more things ‘behind the scenes’ as compared to a classical particle
of precisely defined position and momentum. The result is infinitely richer physics — quantum
physics.