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Meaning does not reside in the mathematical symbols. It resides in the cloud of thought enveloping these symbols. It is conveyed in words; these assign meaning to the symbols.

                                                                                                                    Marvin Chester

The World Of Quantum Mechanics

Quantum mechanics is a mathematical theory that can describe the behavior of objects that are roughly 10,000,000,000 times smaller than a typical human being. Quantum particles move from one point to another as if they are waves. However, at a detector they always appear as discrete lumps of matter. There is no counterpart to this behavior in the world that we perceive with our own senses. One cannot rely on every-day experience to form some kind of  "intuition" of how these objects move. The intuition or "understanding" formed by the study of basic elements of quantum mechanics is essential to grasp the behavior of more complicated quantum systems.

What is Quantum Mechanics?

Quantum mechanics is a theory developed in the early 20th Century to explain the behavior of atoms and other extremely small systems. Initially the theory used the ideas of classical (Newtonian) mechanics to explain several features exhibited by the hydrogen atom. In addition to the rules of classical mechanics, there was one extra rule that was "tacked on". The extra rule said that a certain property, called the "action", of a closed path for a classical particle must be an integer multiple of a constant called "h-bar". Basically, the idea is that if a particle (like an electron in an atom) travels in a circle then that circular path must have an action equal to h-bar, or 2 h-bar, or 3-hbar, etc. This turned out to be sufficient to explain the hydrogen atom, but it could not explain many other atomic phenomena.

Eventually, a new theory was developed that could account for all of the observed atomic phenomena. This theory came in two different forms: one described an atomic system using something called a wavefunction, the other described the system using matrices. The two versions turn out to be equivalent to each other. In addition to providing a mathematical way to describe an atomic system, this new theory also provided a set of rules to determine the behavior of the quantum system in much the same way that Newton's Laws determine the behavior of a classical system. However, this new theory of quantum mechanics is by no means equivalent to Newton's Laws. There are some major differences between classical and quantum mechanics, and these differences are important for our discussion of quantum chaos.

Major Differences Between Newton's Laws and Quantum Mechanics

(A)  In classical mechanics a particle can have any energy and any speed. In quantum mechanics these quantities are quantized. This means that a particle in a quantum system can only have certain values for its energy, and certain values for its speed (or momentum). These special values are called the energy or momentum eigenvalues of the quantum system. Associated with each eigenvalue is a special state called eigenstate. The eigenvalues and eigenstates of a quantum system are the most important features for characterizing that systems behavior. There are no eigenvalues or eigenstates in classical mechanics.

(B)  Quantum mechanics incorporates what is known as the "Heisenberg Uncertainty Principle". This principle states that one cannot know the location AND velocity of a quantum particle to infinite accuracy. The better you know the particle's location, the more uncertain you must be about its velocity, and vice versa. In practice, the level of uncertainty that is required is so small that it is only noticeable when you are dealing with very tiny things like atoms. This is why we cannot see the effects of the Uncertainty Principle in our daily lives.

(C)  Quantum mechanics permits what are called "superpositions of states". This means that a quantum particle can be in two different states at the same time. For instance, a particle can actually be located in two different places at one time. This is certainly not possible in classical mechanics.

(D)  Quantum mechanical systems can exhibit a number of other very interesting features, such as tunneling and entanglement. These features also represent significant differences between classical and quantum mechanics, although they will not be as important in our discussion of quantum chaos.

That is a pretty brief introduction to the ideas of quantum mechanics and many important features have been skipped. But the ideas presented above should make it clear that quantum mechanics is very different from classical (Newtonian) mechanics. We have seen how chaos is defined in classical mechanics. Can chaos also be defined in quantum mechanics? If so, how? We will explore this quesiton in the next section.

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