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.
What is Quantum Mechanics?
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
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
- 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
- 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
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.
[Go To "What is Quantum Chaos?"]
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This website was produced by:
Todd K. Timberlake
Physics, Astronomy, & Geology
If you have comments or suggestions pertaining to this site please email Todd Timberlake.