Download BWilliamsPaper - FSU High Energy Physics

The Road to Quantum Theory
Brian Williams
Around the turn of the twentieth century, our view of the universe was changing
dramatically. With Einstein’s theories of relativity, people were starting to realize that
the universe may not work in ways that was once thought. The intuitive, common-sense
view of the universe built on simple logic and centuries old equations was falling apart at
the seams. Simple problems could no longer be solved by classical methods. Take for
example the problem of calculating the total energy inside an oven. A perfectly
insulated oven at a set temperature will have a finite amount of energy in the form of
electromagnetic radiation. However, physicists using classical methods came up with a
ridiculous answer when they tried to mathematically calculate the total energy, infinity!
Clearly a new theory was required, one that would explain how the oven worked. We
shall return to this problem shortly. But first, we must understand where the laws of
physics at the time had come from.
The ancient Greeks are often the first civilization credited with developing a
scientific method by which to examine the world. It was Eratosthenes who first made a
somewhat accurate measurement of the circumference of the earth, and Aristotle who
first tried to determine what things were made of. He determined that everything on
earth was composed of one of four elements, earth, fire, air, and water. He looked at
these four elements and grouped them into two categories, those that moved up (air and
fire) and those that moved down (water and earth). Aristotle even went so far as to
“discover” a fifth element. He looked at the stars in the sky and noticed that they didn’t
move up or down, so far as he could tell, so he assumed that the stars must be made out
of something else, something he called ether. This crude classification of the elements,
while it lasted a long time, didn’t last forever. About 200 years ago, John Dalton
formulated his atomic theory, which stated that matter is made up of individual units
called atoms, from the Greek word meaning “indivisible.” Furthermore, he proposed
that atoms of one element are different from those of another element. At the time, it
was thought that atoms were the smallest particles possible, the true “building blocks” of
matter. It was later discovered that these atoms are indeed made of smaller parts, with a
heavy nucleus at the center, surrounded by very light electrons in orbit. At first it was
thought that the atom was similar to a mini solar system, with the electrons in definite
orbits around the nucleus. The problem was, it was known in classical mechanics that a
moving charged particle emits radiation, and the electrons would soon spiral down into
the nucleus, and in the span of much less than a second, you wouldn’t have a stable atom.
Clearly something was wrong with this model.
It is also necessary in physics to know the laws governing the behavior of
systems, not merely what they are composed of. Prior to Isaac Newton, a scientist who
was examining an apple falling from a tree might have looked at it and said, “it is in the
nature of apples to fall towards the earth,” and leave it at that. They certainly would
have realized that there was some sort of attractive force between the earth and the apple,
but no one had ever been able to qualitatively describe it. Isaac Newton changed all that.
He not only realized that there was an attractive force between the apple and the earth, he
determined that it was the same attractive force that held the moon in orbit around the
earth, and the earth in orbit around the sun, and so on. He was able to write a
mathematical expression which quantified the force, called gravity, relating the attractive
force between two objects to the product of their masses divided by the square of the
distance between them. Newton also developed three laws of motion, laws which are
still taught in introductory physics courses today. His first law states that a body will
continue in the state of motion (or rest) which it is in until a force is applied to it. His
second law, relating the acceleration an object experiences to the total force applied to it,
is more familiarly known as F=ma. His third law states that for every action, there is an
equal and opposite reaction. It is this law that later lead to principles such as
conservation of momentum. These laws formed the basis of what we now know as
Classical Mechanics. Using mathematical techniques such as algebra, calculus, and
differential equations, we can solve equations of motion for everything from the smallest
particle to the largest galaxy. Newton’s laws held firm for over 200 years before
Einstein’s discoveries would cause us to take a new look at the way the universe works
on large scales. But what about small scales? Could Newton’s laws be applied to
atomic and newly discovered sub-atomic particles, such as the electron? Clearly they
could not. The example of the electron spiraling into the nucleus shows us that
Newton’s laws, while they may work for a real solar system, do not apply to little things.
We shall now return to the oven, which we conveniently left on at a steady
temperature. Classical physicists had tried to calculate the energy in the oven, and came
up with infinities which they could not explain, nor get rid of. Along came Max Planck,
who thought of a new way to look at the oven. At the time, it was thought that all
electromagnetic waves, regardless of their wavelength, carried the same amount of
energy. The amplitude of the wave, it was thought, was what determined how much
energy it had. Make a bigger wave, i.e. one with a bigger amplitude, and you will have a
more energetic wave, in much the same way that a guitar string that is plucked hard is
louder than one that is plucked softly. What Planck suggested was that the energy a
wave could have, or more specifically, the minimum energy a wave could have, was
proportional to its frequency, and therefore its wavelength. Increase the frequency of a
wave, Planck argued, and the minimum amount of energy it can have must go up. So
how does this solve the infinite energy problem of the oven? Planck suggested that
oscillators in the walls were quantized. In other words, the amount of energy that could
be put into the wave could only come in discrete steps, it was not a continuous function.
Planck argued that if the minimum energy carried by a wave was larger than the amount
it was supposed to contribute for that particular oven temperature, that wave could not
contribute to the total energy. By having a finite number of waves with a finite amount
of energy inside the oven, the total energy could only be finite. Planck went on to say
that there was a constant factor which related the energy of a wave to its frequency, a
factor which he called h, which is now known as Planck’s constant, equal to 6.63 *
10^-34 J*s.
There was another problem that had been troubling physicists for hundreds of
years. What was the nature of light, particle or wave? Newton postulated that light was
made up of tiny particles that streamed out from the light source in every direction at the
speed of light. Still other physicists, like Christian Huygens, said that light must be a
wave. In the early 1800’s, Thomas Young set out to determine whether light was a
particle or a wave. He devised a remarkably simple experiment, which today is known
as Young’s double-slit experiment. He used a narrow beam of light coming in from a
window, sent it through a slit, and set up a detector screen to see what came out of the
slit. If light were a particle, he reasoned, it would go straight through the slits with no
interference. However, if there was an interference pattern, then light must be a wave.
Sure enough, an interference pattern was seen on the detector screen, and light was a
wave. We shall return to this double-slit experiment later, it is quite important. Indeed,
Maxwell’s electromagnetic equations were based on the idea that light is a wave.
Fortunately, the particle idea of light was not dead forever. About a hundred years after
Young’s experiment, Einstein decided to re-open the case of the nature of light.
Physicists had discovered that light, when shone on a metal surface, could eject electrons
from the surface. This was not the puzzling part, everyone knew that waves carried
energy, and with energy you should be able to eject electrons. The puzzling part was
that the speed of the electrons being ejected did not depend on the brightness of the light
being shone upon them. If light was a wave, and the intensity of the light was turned up,
then the total amount of energy on the surface of the metal, should increase, and therefore
the amount of kinetic energy each electron had when it was ejected should increase also.
This was not the case. If the experimenter increased the intensity of the light, more
electrons would be ejected, but they would have the same amount of kinetic energy.
Moreover, it was discovered that varying the frequency of the incident light did have a
direct effect on the kinetic energy of the electrons. Einstein looked at this and said that it
could be reconciled if light was again thought of as a particle, with each particle of light
imparting its energy to one electron. Turn up the intensity of the light, and you have
more photons coming in, but since they still have the same individual energy, the amount
of energy they transfer to the electrons will not change. On the other hand, increase the
frequency of the light, and you will increase the energy of the individual photons, and
thus the amount energy they can transfer to the electrons, kicking them off with greater
speed. Einstein would later be awarded the Nobel Prize for this discovery.
Unfortunately, this leaves us right back where we started, wondering if light is a particle
or a wave.
Let us now return to the double-slit experiment discussed earlier. This
experiment has far-reaching results. Richard Feynman, one of quantum mechanics
greatest practitioners, has been quoted as saying that all of quantum mechanics can be
determined from carefully examining the double-slit experiment. We set up a light
source that we can adjust to our specifications, and a detector screen some distance away
from the light source, which will leave a dot whenever a light from the source hits it. In
between we put a barrier, with two slits cut into it, which we can open and close as we
please. We start out by turning on the light source and opening the left slit only. As
one would expect, light shines through the slit and onto the detector screen directly
behind the slit, but nowhere else, since the screen prevents the light from shining
anywhere else. The same thing happens with the right slit. Now we open both slits at
the same time. Now we see an interference pattern, leading us to believe that light is a
wave, and it interferes with itself much the same way waves of water in a tub can
interfere with themselves, causing constructive interference in some places (bright spots),
and destructive interference in others (dark spots). But now Einstein has told us that
light can also be thought of as a particle, due to the photoelectric effect. So how can this
interference pattern be reconciled with the idea that light is a particle. This is easy if we
think of light as being a stream of particles that together, act like a wave, with some
photons canceling each other out, and some photons building on each other.
Unfortunately, this doesn’t work. Let’s say we slow down the source to emit only one
photon at a time, say one every ten seconds. With only the left slit open, we will observe
the exact same pattern as we did before, with the light only being detected directly behind
the slit. The same is true for the right slit. Now we open both slits, with only one
photon in the system at a time. Amazingly, we will observe the same interference
pattern we saw before, with bright spots in some places and dark spots in some places. It
would seem as though the photons somehow know that both slits are open, even though
there is no way they can communicate with each other, since only one is in the system at
any given time. In addition, these individual photons can be detected behind the screen,
a place where they shouldn’t be allowed to be. It is as if the mere fact that the other slit
is open can affect the path taken by a photon to get to the screen. Obviously these
photons that make up light are very different from any particles of matter that we can
observe, or are they?
In 1923, a Frenchman named Louis de Broglie made a prediction about the way
matter should behave. In short, he reasoned that if Einstein’s famous E=mc^2 equation
relates energy and mass, and Planck’s formula relates the energy of a wave to its
wavelength, then matter should have a wave-like property to it as well. He developed a
simple equation which stated that the wavelength of matter was equal to Planck’s
constant divided by the momentum of the body. For large objects, this is
inconsequential, a person, for instance, would have a “de Broglie wavelength” of
something on the order of 10^-36 m, far too small to measure. A sub-atomic particle on
the other hand, with its extremely small momentum, could have a noticeable wavelength.
If electrons, for instance, have a wavelength, then they too should form an interference
pattern when subjected to a double-slit experiment. In the mid-1920’s two physicists at
the Bell laboratories were studying how a stream of electrons interacted with a hunk of
nickel. It turns out that the nickel crystals in this experiment act very much like the slits
in the previous experiment, except that now we are dealing with electrons, which we
know are particles. As it turns out, the results of the experiment are the same, with an
interference pattern detected from the electrons as well. In fact, when the wavelength of
the electrons is calculated from this interference pattern, it agrees with the wavelength
from de Broglie’s hypothesis. This earned de Broglie the Nobel prize in 1929.
Apparently, both waves and light can be thought of as being both waves and particles at
the same time. Depending on the experimental setup, you can see them behaving as
waves or particles. This is not due to a flaw in the experiments, it is simply the way
nature works. It is something we call wave-particle duality, and it’s something
physicists have come to accept about the universe.
Now that we have all these wonderful ideas, energy is emitted only in packets,
matter can be thought of as being a wave, etc., we wish to have a mathematical
interpretation for all of this. After all, no matter how good they may sound, ideas are
fairly useless without mathematical support. In the 1920’s, two physicists, working
independently, came up with two very different mathematical methods to describe
sub-atomic systems. Werner Heisenberg developed matrix mechanics, and Erwin
Schrodinger developed wave mechanics. In Schrodinger’s method, he developed what is
now known as the Schrodinger equation. It took several ideas which were known at the
time, including both de Broglie’s and Planck’s hypotheses, and formulated them into an
equation that also included the wave function, . The Schrodinger equation, when
applied to the hydrogen atom, yielded the same results that physicists such as Bohr had
obtained in other ways. However, there was still the question of what the wave function,
, actually meant. Schrodinger referred to it as a “smeared-out” electron, but this holds
very little physical meaning. What exactly does a smeared-out electron look like? A
pancake? It was Max Born who provided us with a more accurate description of what
the wave function describes. Born said that the wave function describes the particle
from a probability standpoint. Rather than tell you where the particle is, it only tells you
where the particle is most likely to be. The square of the magnitude of the wave
function gives the probability density for a particle over a particular region. If the wave
function is normalized, then the square of the magnitude over a particular interval will
give you the percent chance of finding the particle in that region. This presented a
problem to many physicists of the time, who believed that an equation should be able to
give you an exact answer for where the particle is at a given time, assuming you know the
initial conditions and the outside forces. In fact, this is the biggest break between
classical mechanics and quantum mechanics. In classical mechanics, if one knows the
initial conditions of a particle, and all of the forces acting on the particle, then in principle
you can determine the location of the particle at any time in the future. In quantum
mechanics, however, you can never know exactly where the particle will be in the future,
in fact, you can never know exactly where it is now, a problem that we shall revisit
shortly. Apparently, the quantum world is nothing more than a game a chance, a roll of
the dice, the particle might be here, it might be there, it might be anywhere. This greatly
bothered many physicists of that era. As Einstein said, “God does not play dice with the
universe.” It would seem though, that not only does God play dice, they are loaded dice,
with the probabilities for finding the particle being higher in some places than in others.
We are still left with the problem of having two different mathematical constructs
which both give correct results, matrix mechanics and wave mechanics. This problem
was solved in 1926 by Paul Dirac, who showed that the two were in fact, equal. In doing
so, he also developed a general formalism for quantum mechanics based on linear
algebra, a formalism that we still use today. In 1928, Dirac developed relativistic
quantum mechanics, which incorporated Schrodinger’s ideas, but also explained
something that Schrodinger’s equation did not account for, electron spin, which had just
been discovered a few years earlier. Among the other contributions of Dirac was the
prediction of the existence of antimatter, particularly the prediction of the anti-electron,
or positron, which was discovered experimentally in 1932.
I said before that it was impossible in quantum mechanics to know the initial
conditions of a system exactly. This is due to the Heisenberg Uncertainty Principle,
which states that you cannot simultaneously know the momentum and the position of a
particle with 100% certainty. The reason for this is that a measurement on one variable
will affect the other. For example, let’s say you wanted to know the position of an
electron. The way we determine the position of things is to look at them, which means
that we must shine light on them. But because an electron is so small, even shining a
little photon on it will give it a kick. Once you kick it, it isn’t in the same place
anymore, and it certainly isn’t moving with the same velocity it was before. This crude
example serves our purpose well, because if you hit the electron with a shorter
wavelength photon to get a better determination of its position, you will then give it a
harder kick, since shorter wavelength photons have more energy. The more you know
one variable, the less you can know the other. In fact, this doesn’t just work with
momentum and velocity. It works with any two non-commutable variables, such as
energy and time.
Now that we have a mathematical interpretation of quantum mechanics, let’s look
at a simple problem, the particle in a box problem. Given the wave function of the
particle, we can determine the places where it is most likely to be. If we wish to check
our results we can simply look inside the box to see where the particle is. If we do this
enough times, and graph our results, we should get a curve that looks like the probability
density from the wave function. But let’s say we wanted to know where the particle was
the instant before we opened the box and looked. We already know that there is no way
to solve for this mathematically, since the wave function only gives us statistical
probabilities, but there several interpretations as to where the particle might have been
just before we looked.
1. Realist- The particle was always there. If we opened the box to find it at
position X, then it is logical to assume that it was at X. This was the position that
Einstein, among others, held firmly to. However, if the realist position was correct, then
quantum mechanics was incomplete, since it couldn’t tell us where the particle was,
which was at X. For this reason, hidden variable theories were proposed, with a hidden
variable being something else besides the wave-function that would completely describe
the particle.
2. Orthodox- The particle wasn’t really anywhere. This position was held by
Neils Bohr, among others. This position says that before you opened the box, the
particle was not in a well-defined position. It was everywhere and it was nowhere at the
same time. Perhaps this is what Schrodinger meant when he talked about a smeared-out
electron. This position, as strange as it may sound, has been the most widely held
position among physicists. Now as for why the particle chose position X when you
opened the box is a question that I will not even attempt to discuss here, that is a problem
that physicists are still working on today. Apparently there is something about the act of
making a measurement that causes the wave function to “collapse” into a certain position.
3. Agnostic- Who cares? In a nutshell, this was the position taken by many
physicists when the question arose. We can’t ever know because the question is
metaphysical, you can’t ever answer a question about “before the measurement,” because
to check yourself you would have to perform a measurement, and then you would be
back in the same situation you started in. Wolfgang Pauli advocated this position. He
and other agnostics considered it silly to rack one’s brain over a question such as this.
So which is correct? Theoretical and experimental work in the 1960’s by John
Bell and others showed that apparently, the orthodox position is the correct one. The
particle was not anywhere until you decided to look at it, the act of measurement
collapsed the wave function in a certain place. How and why the wave function decided
to collapse at X, and not some other position, is a question that I am in no position to
answer, indeed, that may very well be a question that we will never know the answer to.
Let’s examine one more paradox that was posed in the early years of quantum
theory, that of Schrodinger’s Cat. The following is taken from an essay written by
Schrodinger in an essay written in 1935. “A cat is placed in a steel chamber, together
with the following hellish contraption...In a Geiger counter there is a tiny amount of
radioactive substance, so tiny that maybe within an hour one of the atoms decays, but
equally probably none of them decays. If one decays then the counter triggers and via a
relay activates a little hammer which breaks a container of cyanide. If one has left this
entire system for over an hour, then one would say the cat is living if no atom is decayed.
The first decay would have poisoned it. The wave function of the entire system would
express this by containing equal parts of the living and dead cat.” In other words, until
you looked in the box, the cat would not be in a definite state of either living or dead, it
would be in a “smeared-out” state of both dead and alive, only by opening the box would
the cat be forced to take a stand on which position it wanted to take. If this seems like
nonsense, it is. Schrodinger knew this, in fact, the question was posed as a ridiculous
argument to show how silly he thought the orthodox position was. Obviously the cat is
either alive or dead, there is no such thing as a quasi-alive state for cats (or any other
organism for that matter). It appears as though the Geiger counter in the box acts as the
measuring device, causing the wave function of the system to collapse. The cat example
does serve one important purpose, however, it shows us that quantum mechanical
principles do not apply on large scales.
This presents somewhat of a problem. In the last 100 years, we have developed
two wonderful theories that explain their respective areas very well. Quantum
mechanics works quite well on small scales, but does not apply to large scales, whereas
General Relativity works wonderfully on large scales such as the universe itself, but it
cannot explain the strange world of sub-atomic particles. The search is on for a grand
theory that will unite these two theories into one mathematical and theoretical description
that will work on both scales. This unified theory, if it exists, is the holy grail of modern
physics.
Bibliography
Greene, Brian. The Elegant Universe. New York: Vintage Books, 1999.
Griffiths, David. Introduction to Quantum Mechanics. Upper Saddle River, NJ:
Prentice
Hall, 1995.
Ferris, Timothy. Coming of age in the Milky Way. New York: Anchor Books, 1988.
Oxford Dictionary of Scientists. Ed. John Daintith and Derek Gjertsen. Oxford
University Press, 1999.
Schrodinger, Erwin, Naturwiss. 48, 52 (1935)