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**

- 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 an__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. - Newton's Laws allow one, in principle, to determine the exact
location and velocity of a particle at some future time. Quantum
mechanics, on the other hand, only determines the
__probability__for a particle to be in a certain location with a certain velocity at some future time. The probabilistic nature of quantum mechanics makes it very different from classical mechanics. - 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. - 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.
- 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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This website was produced by:

Todd K. Timberlake

Assistant Professor

Department of Physics, Astronomy, & Geology

Berry College

If you have comments or suggestions pertaining to this site please email Todd Timberlake.