In classical mechanics, the state of a system may be specified by
giving the coordinates and velocities of all
of the particles in the system. Once this is done, all measurable
quantities such as the momenta , total energy and angular momentum of the system may be determined. Such
measurable quantities are known as * observables*. Under the classical
principle of determinism, once the state of the system is specified at
time **t = 0**, the laws of motion may be solved to determine the state
of the system for all later times.

The situation in quantum mechanics is not much different. We may
still make measurements of the familiar physical observables mentioned
above (and some new ones such as spin or polarization which we will
learn about later). As we have seen in our discussions of the
uncertainty principle and interference experiments, although the
results of * individual* measurements are unpredictable on the
quantum scale, measurements on a system in a given state yield a
well-defined * distribution* of results. If we simply accept this
notion and generalize our concept of ``experiment" to the procedure of
determining the distribution associated with many measurements of the
same observable over a set of identically prepared systems, then we
may maintain the idea that the * quantum state* of a system
determines the results of all experiments measuring physical
observables.

As we shall refer to the quantum state of a system often, we shall
make a special symbol for it, . The symbol
is called a ``ket.'' We may place different symbols inside
the ket to distinguish different states. For instance, our system may
be in the state at time **t = 0** and in the state
at some later time.

Keep in mind that, in general, does not determine
the result of any * individual* measurement of an observable, only
probability distributions such as , the probability of
finding a particle with position **x** in the range , or
, the probability of finding the momentum of the
particle in the range . In the case of photons we may
also consider the observable of spin or polarization and measure , the probability of observing a right or
left circularly polarized photon.

We mention the polarization of photons above to underscore the fact that we will keep the abstract portion of our discussion completely general. For the most part of the course, however, we will focus on systems of single particles which have no internal structure so that we will concern ourselves with only the observables of position , momentum and those observables which may be derived from these two, such as energy and angular momentum .

Wed Oct 11 21:10:55 EDT 1995