In section (2), we found inner product expressions for the average position <x> and momentum <p> in terms of the quantum amplitudes or ``wave functions'' in either the position or momentum representation. In all cases we found an expression of the form
where 
is the physical observable begin averaged, 
are
the quantum amplitudes in the representation and 
is a
function operator designed to give the correct average when used in
the corresponding representation. Here the double over-bar ``
'' refers to a generic representation which could either be
position (for which we usually give no over-symbol) or momentum (for
which we usually give a tilde `` ''). With equation (14)
and knowledge of the correct operator, we may compute the average of
position or momentum in either the position or the momentum
representation. The table below summarizes the operators we have
found so far:
TABLE III: OPERATORS DERIVED IN THIS NOTE
As we shall see below, in each and every representation
, we may find a (unique) operator 
associate with each and every observable 
of a system so that the
form (14) is valid. We call this the physical
operator associated with the observable, and represent the operator
as the symbol for the observable (e.g., x for position, p for
momentum, 
for the z component of angular momentum, H for the
energy, or 
for a generic operator, et cetera) with an
over-symbol telling the appropriate representation (e.g., no symbol
for the position representation, an over-tilde ``
'' for
the momentum representation, an over-bar ``
'' for the
representation associated with pure states of the observable itself
and a double over-bar ``
'' for a generic, unspecified
representation) with a final caret or ``hat'' (``
'') on top
to remind us that we are dealing with a function operator.
Definition: The operator associated with an observable in a given
representation is constructed so that averages of the observable are
always given by 
, We
refer to all operators defined with respect to physical observables in
this way as physical or observable operators.
All operators in quantum mechanics are either associated directly with
physical observables or are constructed from such physical operators
using the algebraic rules laid out in section (3). It is
thus important to understand the special properties of physical
operators. This is the purpose of this section.
To see how the average of any observable in any representation takes the generic form
we begin by considering the form for the average in the representation of pure states of the associated observable. In this representation, the average clearly has the above form. We will then argue that the above form stays invariant as we move from representation to representation so that the form above is completely general.
To determine the form of operators in their pure state representation,
we begin with the principle of superposition and represent the
state 
as a combination of pure states of the observable
,
Here
are the quantum amplitudes (wave function) for the
state in the 
representation so that the probability of measuring
the value 
of the observable in an experiment is
. Thus, in the pure representation of
any observable,
We see that in the pure representation we always have that the operator associated with the observable looks like argument multiplication,
(We have
already seen two examples of this, 
and 
.)
To transform our expression (15) for general averages
to other representations, we write 
and apply
Parseval's theorem as we did in section (2),
where the operation of 
on 
proceeds in
three well-defined mathematical steps whose application in sequence
defines the operator 
: 1) transform 
to the pure
state representation to generate 
, 2) multiply 
by 
to make 
, and 3) transform 
back to the initial representation to get 
. Although the procedure defining 
is
complicated and involved, it may be applied to any function 
as input and will yield a single 
as
output. This procedure thus qualifies as a valid definition of an
operator. As we have seen with position and momentum,
although the defining procedure is involved, it often results in a
simple final operation on 
to yield 
. In the
following sections we
shall see further that this definition results in some simple buy very
important and general properties for the operators we will encounter
in quantum mechanics.
