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
