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Quantum mechanics is the theory that we use to describe the microscopic world. The
microscopic world is the realm of atoms, photons, nuclei, electrons, neutrons, and a
whole host of other subatomic particles. These particles are the “building blocks” of
our universe, in the sense that everything that is observable in the macroscopic world
that we can see and feel around us is the result of the presence and interactions of
these elementary particles.
Quantum mechanics is also an inherently probabilistic theory, in that there exists
uncertainty at the most fundamental level when we try to measure any value, or
observable, of a system. This is unlike classical and relativistic theories, where
everything exists with precise and definite values, and the time evolution of a system
can theoretically be determined as far into the future as we want. This is not the case
in quantum mechanics; we can only define probabilities to the future behaviour of a
system. Despite its seemingly limited nature, quantum mechanics has been extremely
successful at explaining and illuminating the microscopic world, which classical
Newtonian mechanical theory is unsuccessful at explaining.
First, a little history. The initial seed that was to later grow into the quantum theory
that we have today was Max Planck’s postulate that energy is quantised, or comes in
discrete packages, as opposed to existing in an infinitely continuous series of states.
He put forward this postulate in order to explain the phenomena of blackbody
radiation. He postulated that an electromagnetic wave can only interact with matter in
integer multiples of h, where  is the frequency of the wave, and h is a quantity
known as Planck’s constant. Planck’s constant has a value of h = 1.05457  10-34 Js,
which is an extremely small number, and it is because the value of Planck’s constant
is so small that we don’t ordinarily notice any of the behaviour associated with
quantum mechanics in the macroscopic world around us. Only by probing the
microscopic world do we encounter the strange and mystifying behaviour of subatomic particles.
In 1905, Albert Einstein used the idea of quantised states to explain the photoelectric
effect. He explained the observed frequency dependence of the emitted particles by
postulating that light energy too comes in tiny discrete bits, or quanta. These discrete
light packets, or photons, also come in integer multiples of h, with  here being the
frequency of the light.
A paradigm shift was beginning. In 1913 Neils Bohr was able to explain the observed
spectral lines of a hydrogen atom by quantising the atom. He postulated that atoms
could only exists in states of discrete energy, and that any interaction between a
hydrogen atom and any surrounding radiation could only happen as integer multiples
of h. For example, the hydrogen atom can only be raised to the first excited energy
state from its ground state if it absorbs exactly one times h. Likewise it can only be
raised to it’s second excited state from it’s ground state if it absorbs exactly two times
h, and so on. Bohr’s quantisation of the hydrogen atom successfully explained the
problems that the classical model could not.
In 1924, Louis De Broglie showed that matter itself had wavelike properties. He
derived the equation,  = h/p, that related a particle’s momentum with an associated
wavelength. This equation tells us that all matter has wavelike properties, and must in
some cases be thought of as existing as a wave, rather than as a discrete particle.
The explanations put forward by these scientists successfully explained many
outstanding problems, and only by assuming quantised states. However these
explanations were not based on any underlying theory, with the idea of quantisation
seeming to be arbitrarily postulated. A more rigorous theory was needed, and soon
emerged, extending and developing the work of Planck, Einstein, Bohr, De Broglie,
and others. This theory came to be known as quantum mechanics.
In 1927, Werner Heisenberg developed his uncertainty principle, which states that
you cannot know both the momentum and position of a particle at the same time with
absolute certainty. The uncertainty principle gives us a limit to our knowledge of a
system, and tells us that we can never determine exactly the future behaviour of a
system, because we can never exactly determine it’s present state! In more
mathematical terms, we can never determine two related observables simultaneously.
We can only say that something might happen with some probability, but never with
certainty.
The Schrodinger equation is another fundamental part of quantum mechanics. This
wave equation, which Schrodinger derived in 1925, tells us the evolution of a
quantum mechanical system in time. The full equation is:
[-/2m2 + V(r)](r,t) = i(r,t)/t
This equation is a classical equation in the sense that it does not take into account
special relativity. It has the classical kinetic energy term on the left, rather than the
relativistic term for kinetic energy. Paul Dirac was to derive a relativistic version of
the Schrodinger equation in 1928.
In 1925 the concept of spin was introduced by Ralph Kronig. Spin is the intrinsic
angular momentum that a microscopic particle possesses, although the concept of spin
is a bit different from what it means in classical mechanics. In classical mechanics, an
objects angular momentum is due to its rotation around its central axis, or around an
extended axis.
Spin angular momentum in quantum mechanics does not arise from a particle actually
spinning like a top, rather it is an intrinsic property of a particle, like its mass. An
important thing to note is that spin is quantised. It can only have discrete values. For
example, protons, neutrons and electrons are all “spin half” particles, that is, they
have spin values that are one half , where  = h/2, and h = Planck’s constant. A
hypothetical particle that will play a large part in the conflict between general
relativity and quantum mechanics is the graviton. This as yet undetected particle is a
spin two particle, meaning two times .
Let’s look at what quantum mechanics has to say about the fundamental forces. We
have known since Newton’s time what forces are. A force can basically be defined as
something that causes the state of an object to change; whether it is a change in the
object’s motion, temperature, electrical charge, or potential energy; a force is
responsible. However Newtonian mechanics does not specify any mechanism by
which force is transferred, except in the most basic sense. It tells us that when an
object is in contact with another you have some force between them. Newton’s laws
tell you what the effect of a force is, and how to calculate the magnitude of a force,
but they do not have much to say about what actually happens at the microscopic
level when forces act - the actual mechanism by which a force acts.
Quantum mechanics gives us that mechanism. It has been determined after much
smashing of particles together that forces are the result of the exchange of force
particles that transfer force from one object to another. Every interaction in the
known universe can be explained by the exchange and effect of force particles. A
wonderful discovery relating to forces has been that all interactions between objects
can be reduced to the work of four fundamental forces. Those fundamental forces are:
The strong nuclear force, the weak nuclear force, the electromagnetic force, and
gravity. And each of these four forces has an associated force particle that is
responsible for the force.
The strong nuclear force is the force that holds together protons and neutrons in an
atomic nucleus. More specifically, the strong force can be thought of as having two
aspects, one which holds together the quarks within protons and neutrons, and the
other that holds protons and neutrons together in a nucleus, known as the fundamental
strong force and the residual strong force respectively. The fundamental strong force
arises due to the interactions of eight different types of force particles called gluons.
The residual strong force is due to the interactions of mesons. The strong nuclear
force is a very short range force, not usually having any influence further than the
radius of a proton or neutron.
The weak nuclear force is responsible for things like radioactive beta decay, which is
when a neutron turns into a proton, with an electron and an anti-neutrino being
emitted in the process. The force particles responsible for this force are called W and
Z bosons. The weak nuclear force is also a very short range force, only acting over
distances not bigger than an atomic nucleus. This force is about one billion times
weaker than the strong force.
The electromagnetic force is the force that acts between electrically charged particles.
This includes all electric and magnetic forces which arise from the motion of charged
particles, and also from stationary electric charges. This force is responsible for most
of the phenomena we see around us, such as light, friction, and the structure of
elements and molecules. This force can be both attractive and repulsive. The force
particle responsible for electromagnetic interactions is the photon.
When we say that the interactions of forces are due to the exchange of force particles,
this means that force particles are emitted from their “parent” particles, and these
force particles then interact on another particle, and this interaction results in a force.
To use the electromagnetic force as an example, we can think of an electromagnetic
field consisting of huge cloud of photons that are emitted from a charged particle.
When two charged particles exert force on one another, they shoot photons out
between themselves. Classically this looks like two electromagnetic fields interacting.
The fourth fundamental force is Gravity. It is the attraction that all masses have for
each other. It is the weakest of the four forces, approximately 1036 times weaker than
the electromagnetic force, but it is always attractive, and has the longest range of all
the forces. Consistency would indicate that a force particle must also exist for the
force of gravity, and has been dubbed the graviton. Since the other three forces have
been successfully explained by assuming force particles, it makes sense that gravity
would also be the result of some particle exchange. So the graviton was postulated,
with the expectation that a quantum gravity theory would quickly pop out, one that is
consistent with general relativity.
However this was not to be. Conflicts and inconsistencies soon arose due to the
fundamental differences between relativity and quantum mechanics, which will be
elaborated on in later pages.
One fundamental difference is that general relativity says that spacetime is warped
due to the presence of masses, and the force of gravity is not the result of any particle,
but due to the curvature of spacetime. For example, a planet circling the sun appears
to be in orbit due to the gravitational force exerted by the sun, but general relativity
tells us that the planet’s orbit is due to its passage through curved space. It is
attempting to take the straightest path through curved geometry. Thus no force
particle is needed in general relativity. The nature of spacetime will be explained in
the general relativity section of this website.
An interesting thing happens when we try to probe the microscopic world at greater
and greater levels of magnification; space becomes more and more turbulent, or
frothy. Space loses its “smoothness” in that quantum indeterminancy becomes more
and more prominent. The Heisenberg uncertainty principle gives us a limit as to how
much we can know about a system. The more we try to pin down space and time, the
greater the indeterminancy in the system grows, in that the values a system can take
grow larger and larger – the values can fluctuate wildly. As we zoom in on space and
time, we find that quantum fluctuations start to turn space and time into a violent and
energetic proto-stuff, or as the physicist John Wheeler calls it, “quantum foam”.
Einstein’s smooth spacetime soon becomes a hyperactive froth where concepts such
as up-down, left-right, before-after, cease to have any meaning. When we try to
examine the gravitational field, we find that the quantum fluctuations distort and warp
space and time so much that it is simply impossible to think of space and time as
smooth. The weakness of gravity and the miniscule size of Planck’s constant combine
to give us a value to where these fluctuations take place, known as the Planck length,
which is roughly 10-35 metres in length. In this realm of existence, the geometry of
spacetime is smashed up and destroyed by the Heisenberg uncertainty principle.
Quantum field theory is the theory that attempts to unify special relativity and
quantum mechanics. Quantum field theory is necessary because the schrodinger
equation is a non-relativistic equation, in that it reduces to classical mechanics instead
of relativistic mechanics in the correspondence limit.
In quantum mechanics time is thought to have a fixed, non-dynamic structure as
opposed to relativity. Time exists as an absolute quantity, like in Newtonian
mechanics; it is the background on which quantum mechanical interactions take
place. However just as there is a limit in our measurements of space due to the
Heisenberg uncertainty principle, so is there a limit when we try to measure time. The
smallest unit of time that has any meaning in quantum mechanics is known as Planck
time, which has a value of: 5.391  10-44 seconds. This is the length of time it would
take a photon travelling at the speed of light to travel one Planck length. Also, when
we try to examine the origin of the universe, we cannot determine or measure any
difference between the actual moment the universe came into existence and one
Planck time unit after.
With the concepts and ideas introduced in the general relativity section and quantum
mechanics section, we are now ready to examine the conflict between the two theories
more closely, and look at the possible solutions.