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 
.