The result of a measurement of the observable *A* must yield one of the
eigenvalues of *A*. Thus, we see why *A* is required to be a hermitian
operator: Hermitian operators have *real* eigenvalues. If we
denote the set of eigenvalues of *A* by , then each of
the eigenvalues satisfies an eigenvalue equation

where is the corresponding eigenvector. Since the operator *A* is
hermitian and is therefore real, we have also the left eigenvalue equation

The probability amplitude that a measurement of *A* will yield the eigenvalue
is obtained by taking the inner product of the corresponding eigenvector
with the state vector , . Thus, the probability that
the value is obtained is given by

Another useful and important property of hermitian operators is that their eigenvectors form a complete orthonormal basis of the Hilbert space, when the eigenvalue spectrum is non-degenerate. That is, they are linearly independent, span the space, satisfy the orthonormality condition

and thus any arbitrary vector can be expanded as a linear combination of these vectors:

By multiplying both sides of this equation by and using the orthonormality condition, it can be seen that the expansion coefficients are

The eigenvectors also satisfy a closure relation:

where *I* is the identity operator.

Averaging over many individual measurements of *A* gives rise to an
average value or expectation value for the observable *A*, which
we denote and is given by

That this is true can be seen by expanding the state vector in
the eigenvectors of *A*:

where are the amplitudes for obtaining the eigenvalue
upon measuring *A*, i.e., . Introducing this
expansion into the expectation value expression gives

The interpretation of the above result is that the expectation value of *A* is the
sum over possible outcomes of a measurement of *A* weighted by the
probability that each result is obtained. Since is this probability, the equivalence of the
expressions can be seen.

Two observables are said to be compatible if *AB*=*BA*. If this is true,
then the observables can be diagonalized simultaneously to yield the
same set of eigenvectors. To see this, consider the action of *BA* on
an eigenvector of *A*. . But
if this must equal , then the only way this can be true is
if yields a vector proportional to which means
it must also be an eigenvector of *B*. The condition *AB*=*BA* can be
expressed as

where, in the second line, the quantity is know as the
commutator between *A* and *B*. If [*A*,*B*]=0, then *A* and *B* are said to
commute with each other. That they can be simultaneously diagonalized implies
that one can simultaneously predict the observables *A* and *B* with the
same measurement.

As we have seen, classical observables are functions of position *x* and
momentum *p* (for a one-particle system). Quantum analogs of classical
observables are, therefore, functions of the operators *X* and *P*
corresponding to position and momentum. Like other observables
*X* and *P* are linear hermitian operators. The corresponding eigenvalues
*x* and *p* and eigenvectors and satisfy the
equations

which, in general, could constitute a *continuous* spectrum of eigenvalues
and eigenvectors. The operators *X* and *P* are not compatible.
In accordance with the Heisenberg uncertainty principle (to be discussed
below), the commutator between
*X* and *P* is given by

and that the inner product between eigenvectors of *X* and *P* is

Since, in general, the eigenvalues and eigenvectors of *X* and *P* form
a continuous spectrum, we write the orthonormality and closure relations
for the eigenvectors as:

The probability that a measurement of the operator *X* will yield an eigenvalue *x*
in a region *dx* about some point is

The object is best represented by a continuous function
often referred to as the *wave function*. It is a representation
of the inner product between eigenvectors of *X* with the state vector.
To determine the action of the operator *X* on the state vector in the
basis set of the operator *X*, we compute

The action of *P* on the state vector in the basis of the *X* operator is consequential
of the incompatibility of *X* and *P* and is given by

Thus, in general, for any observable *A*(*X*,*P*), its action on the state
vector represented in the basis of *X* is

Wed Mar 10 13:14:21 EST 1999