To them, I said,
the truth would be literally nothing
but the shadows of the images.
-Plato summarized many of the things which he had learned from his teacher, the philosopher Socrates. One of the most famous of these Dialogues is the `Allegory of the Cave'. In this allegory, people are chained in a cave so that they can only see the shadows which are cast on the walls of the cave by a fire. To these people, the shadows represent the totality of their existence - it is impossible for them to imagine a reality which consists of anything other than the fuzzy shadows on the wall.

However, some prisoners may escape from the cave; they may go out into the light of the sun and behold true reality. When they try to go back into the cave and tell the other captives the truth, they are mocked as madmen.

Of course, to Plato this story was just meant to symbolize mankind's struggle to reach enlightenment and understanding through reasoning and open-mindedness. We are all initially prisoners and the tangible world is our cave. Just as some prisoners may escape out into the sun, so may some people amass knowledge and ascend into the light of true reality.

What is equally interesting is the literal interpretation of Plato's tale: The idea that reality could be represented completely as `shadows' on the walls.

In 1993 the famous Dutch theoretical physicist G. 't Hooft put forward a bold proposal which is reminiscent of Plato's Allegory of the Cave. This proposal, which is known as the Holographic Principle, consists of two basic assertions:

Assertion 1 The first assertion of the Holographic Principle is that all of the information contained in some region of space can be represented as a `Hologram' - a theory which `lives' on the boundary of that region. For example, if the region of space in question is the DAMTP Tearoom, then the holographic principle asserts that all of the physics which takes place in the DAMTP Tearoom can be represented by a theory which is defined on the walls of the Tearoom.

Assertion 2 The second assertion of the Holographic Principle is that the theory on the boundary of the region of space in question should contain at most one degree of freedom per Planck area.

A Planck area is the area enclosed by a little square which has side length equal to the Planck length, a basic unit of length which is usually denoted L_p. The Planck length is a fundamental unit of length, because it is the parameter with the dimensions of length which can be constructed out of the basic constants G (Newton's constant for the strength of gravitational interactions), (Planck's constant from quantum mechanics), and c (the speed of light). A quick calculation reveals that L_p is very small indeed:

To many people, the Holographic Principle seems strange and counterintuitive: How could all of the physics which takes place in a given room be equivalent to some physics defined on the walls of the room? Could all of the information contained in your body actually be represented by your `shadow'?

In fact, the way in which the Holographic Principle appears in M-theory is much more subtle. In M-theory we are the shadows on the wall. The `room' is some larger, five-dimensional spacetime and our four-dimensional world is just the boundary of this larger space. If we try to move away from the wall, we are moving into an extra dimension of space - a fifth dimension. In fact, people have recently been trying to think of ways in which we might actually experimentally `probe' this fifth dimension.

At the heart of many of these exciting ideas is a version of the Holographic Principle known as the adS/CFT correspondence.

The adS/CFT correspondence is a type of duality, which states that two apparently distinct physical theories are actually equivalent. On one side of this duality is the physics of gravity in a spacetime known as anti-de Sitter space (adS). Five-dimensional anti-de Sitter space has a boundary which is four-dimensional, and in a certain limit looks like flat spacetime with one time and three space directions. The adS/CFT correspondence states that the physics of gravity in five-dimensional anti-de Sitter space, is equivalent to a certain supersymmetric Yang-Mills theory which is defined on the boundary of adS. This Yang-Mills theory is thus a `hologram' of the physics which is happening in five dimensions. The Yang-Mills theory has gauge group SU(N), where N is very large, and it is said to be `supersymmetric' because it has a symmetry which allows you to exchange bosons and fermions. The hope is that this theory will eventually teach us something about QCD (quantum chromodynamics), which is a gauge theory with gauge group SU(3). QCD describes interactions between quarks. However, QCD has much less symmetry than the theory defined on the boundary of adS; for example, QCD has no supersymmetry. Furthermore, we still don't know how to incorporate a crucial property of QCD, known as asymptotic freedom.

If the above picture is your idea of an atom, with electrons looping around the nucleus, you are about 70 years out of date. It's time to open your eyes to the modern world of quantum mechanics! The picture below shows some plots of where you would most likely find an electron in a hydrogen atom (the nucleus is at the center of each plot).

What is quantum mechanics?

Simply put, quantum mechanics is the study of matter and radiation at an atomic level.

Why was quantum mechanics developed?

In the early 20th century some experiments produced results which could not be explained by classical physics (the science developed by Galileo Galilei, Isaac Newton, etc.). For instance, it was well known that electrons orbited the nucleus of an atom. However, if they did so in a manner which resembled the planets orbiting the sun, classical physics predicted that the electrons would spiral in and crash into the nucleus within a fraction of a second. Obviously that doesn't happen, or life as we know it would not exist. (Chemistry depends upon the interaction of the electrons in atoms, and life depends upon chemistry). That incorrect prediction, along with some other experiments that classical physics could not explain, showed scientists that something new was needed to explain science at the atomic level.

If classical physics is wrong, why do we still use it?

Classical physics is a flawed theory, but it is only dramatically flawed when dealing with the very small (atomic size, where quantum mechanics is used) or the very fast (near the speed of light, where relativity takes over). For everyday things, which are much larger than atoms and much slower than the speed of light, classical physics does an excellent job. Plus, it is much easier to use than either quantum mechanics or relativity (each of which require an extensive amount of math).

What is the importance of quantum mechanics?

The following are among the most important things which quantum mechanics can describe while classical physics cannot:

• Discreteness of energy
• The wave-particle duality of light and matter
• Quantum tunneling
• The Heisenberg uncertainty principle
• Spin of a particle

Discreteness of energy

If you look at the spectrum of light emitted by energetic atoms (such as the orange-yellow light from sodium vapor street lights, or the blue-white light from mercury vapor lamps) you will notice that it is composed of individual lines of different colors. These lines represent the discrete energy levels of the electrons in those excited atoms. When an electron in a high energy state jumps down to a lower one, the atom emits a photon of light which corresponds to the exact energy difference of those two levels (conservation of energy). The bigger the energy difference, the more energetic the photon will be, and the closer its color will be to the violet end of the spectrum. If electrons were not restricted to discrete energy levels, the spectrum from an excited atom would be a continuous spread of colors from red to violet with no individual lines.

The concept of discrete energy levels can be demonstrated with a 3-way light bulb. A 40/75/115 watt bulb can only shine light at those three wattage's, and when you switch from one setting to the next, the power immediately jumps to the new setting instead of just gradually increasing.

It is the fact that electrons can only exist at discrete energy levels which prevents them from spiraling into the nucleus, as classical physics predicts. And it is this quantization of energy, along with some other atomic properties that are quantized, which gives quantum mechanics its name.

The wave-particle duality of light and matter

In 1690 Christiaan Huygens theorized that light was composed of waves, while in 1704 Isaac Newton explained that light was made of tiny particles. Experiments supported each of their theories. However, neither a completely-particle theory nor a completely-wave theory could explain all of the phenomena associated with light! So scientists began to think of light as both a particle and a wave. In 1923 Louis de Broglie hypothesized that a material particle could also exhibit wavelike properties, and in 1927 it was shown (by Davisson and Germer) that electrons can indeed behave like waves.

How can something be both a particle and a wave at the same time? For one thing, it is incorrect to think of light as a stream of particles moving up and down in a wavelike manner. Actually, light and matter exist as particles; what behaves like a wave is the probability of where that particle will be. The reason light sometimes appears to act as a wave is because we are noticing the accumulation of many of the light particles distributed over the probabilities of where each particle could be.

For instance, suppose we had a dart-throwing machine that had a 5% chance of hitting the bulls-eye and a 95% chance of hitting the outer ring and no chance of hitting any other place on the dart board. Now, suppose we let the machine throw 100 darts, keeping all of them stuck in the board. We can see each individual dart (so we know they behave like a particle) but we can also see a pattern on the board of a large ring of darts surrounding a small cluster in the middle. This pattern is the accumulation of the individual darts over the probabilities of where each dart could have landed, and represents the 'wavelike' behavior of the darts. Get it?

Quantum tunneling

This is one of the most interesting phenomena to arise from quantum mechanics; without it computer chips would not exist, and a 'personal' computer would probably take up an entire room. As stated above, a wave determines the probability of where a particle will be. When that probability wave encounters an energy barrier most of the wave will be reflected back, but a small portion of it will 'leak' into the barrier. If the barrier is small enough, the wave that leaked through will continue on the other side of it. Even though the particle doesn't have enough energy to get over the barrier, there is still a small probability that it can 'tunnel' through it!

Let's say you are throwing a rubber ball against a wall. You know you don't have enough energy to throw it through the wall, so you always expect it to bounce back. Quantum mechanics, however, says that there is a small probability that the ball could go right through the wall (without damaging the wall) and continue its flight on the other side! With something as large as a rubber ball, though, that probability is so small that you could throw the ball for billions of years and never see it go through the wall. But with something as tiny as an electron, tunneling is an everyday occurrence.

On the flip side of tunneling, when a particle encounters a drop in energy there is a small probability that it will be reflected. In other words, if you were rolling a marble off a flat level table, there is a small chance that when the marble reached the edge it would bounce back instead of dropping to the floor! Again, for something as large as a marble you'll probably never see something like that happen, but for photons (the massless particles of light) it is a very real occurrence.

The Heisenberg uncertainty principle

People are familiar with measuring things in the macroscopic world around them. Someone pulls out a tape measure and determines the length of a table. A state trooper aims his radar gun at a car and knows what direction the car is traveling, as well as how fast. They get the information they want and don't worry whether the measurement itself has changed what they were measuring. After all, what would be the sense in determining that a table is 80 cm long if the very act of measuring it changed its length!

At the atomic scale of quantum mechanics, however, measurement becomes a very delicate process. Let's say you want to find out where an electron is and where it is going (that trooper has a feeling that any electron he catches will be going faster than the local speed limit). How would you do it? Get a super high powered magnifier and look for it? The very act of looking depends upon light, which is made of photons, and these photons could have enough momentum that once they hit the electron they would change its course! It's like rolling the cue ball across a billiard table and trying to discover where it is going by bouncing the 8-ball off of it; by making the measurement with the 8-ball you have certainly altered the course of the cue ball. You may have discovered where the cue ball was, but now have no idea of where it is going (because you were measuring with the 8-ball instead of actually looking at the table).

Werner Heisenberg was the first to realize that certain pairs of measurements have an intrinsic uncertainty associated with them. For instance, if you have a very good idea of where something is located, then, to a certain degree, you must have a poor idea of how fast it is moving or in what direction. We don't notice this in everyday life because any inherent uncertainty from Heisenberg's principle is well within the acceptable accuracy we desire. For example, you may see a parked car and think you know exactly where it is and exactly how fast it is moving. But would you really know those things exactly? If you were to measure the position of the car to an accuracy of a billionth of a billionth of a centimeter, you would be trying to measure the positions of the individual atoms which make up the car, and those atoms would be jiggling around just because the temperature of the car was above absolute zero!

Heisenberg's uncertainty principle completely flies in the face of classical physics. After all, the very foundation of science is the ability to measure things accurately, and now quantum mechanics is saying that it's impossible to get those measurements exact! But the Heisenberg uncertainty principle is a fact of nature, and it would be impossible to build a measuring device which could get around it.

Spin of a particle

In 1922 Otto Stern and Walther Gerlach performed an experiment whose results could not be explained by classical physics. Their experiment indicated that atomic particles possess an intrinsic angular momentum, or spin, and that this spin is quantized (that is, it can only have certain discrete values). Spin is a completely quantum mechanical property of a particle and cannot be explained in any way by classical physics.

It is important to realize that the spin of an atomic particle is not a measure of how it is spinning! In fact, it is impossible to tell whether something as small as an electron is spinning at all! The word 'spin' is just a convenient way of talking about the intrinsic angular momentum of a particle.

Magnetic resonance imaging (MRI) uses the fact that under certain conditions the spin of hydrogen nuclei can be 'flipped' from one state to another. By measuring the location of these flips, a picture can be formed of where the hydrogen atoms (mainly as a part of water) are in a body. Since tumors tend to have a different water concentration from the surrounding tissue, they would stand out in such a picture.

What is the Schrödinger equation?

Every quantum particle is characterized by a wave function. In 1925 Erwin Schrödinger developed the differential equation which describes the evolution of those wave functions. By using Schrödinger's equation scientists can find the wave function which solves a particular problem in quantum mechanics. Unfortunately, it is usually impossible to find an exact solution to the equation, so certain assumptions are used in order to obtain an approximate answer for the particular problem.

What is a wave packet?

As mentioned earlier, the Schrödinger equation for a particular problem cannot always be solved exactly. However, when there is no force acting upon a particle its potential energy is zero and the Schrödinger equation for the particle can be exactly solved. The solution to this 'free' particle is something known as a wave packet (which initially looks just like a Gaussian bell curve). Wave packets, therefore, can provide a useful way to find approximate solutions to problems which otherwise could not be easily solved.

Doomed Star Eta Carinae

Even though Eta Carinae is more than 8,000 light-years away, features 10 billion miles across (about the diameter of our solar system) can be distinguished. Eta Carinae suffered giant outburst about 150 years ago, when it became one of the brightest stars in the southern sky. Though the star released as much visible light as a supernova explosion, it survived the outburst. Somehow, the explosion produced two lobes and a large, thin equatorial disk, all moving outward at about 1.5 million miles per hour. Estimated to be 100 times heftier than our Sun, Eta Carinae may be one of the most massive stars in our galaxy.