In each of the four cases presented in (3) and
(4), we find that the average of an observable is computed
as the inner product of the wave function with some other function
derived from the wave function through some mathematical operation.
We will see in section (4) that this structure for the
calculation of averages is *
completely general* in quantum mechanics.

In this section we will focus on the mathematics of operations
performed on functions to produce other functions. We will call such
an operation a * function operator*. A function operator is a
generalization of our familiar notion of a function. We normally
think of a function as a ``map'' which associates a single real number
as output for every real number given to the function as input.
However, there is nothing in the idea of a function that requires its
input and output to be real numbers. They could be complex numbers or
the input and output could be functions themselves. A function
operator is then just a special case of this general notion of
function. A function operator accepts functions as input, and
maps each input function to a single unique function as output.
We call these special functions ``function'' operators to remind us
that as input and output, they accept and give functions rather than
numbers. Later, in section (4) we shall define *
quantum* operators, which map not functions but * quantum states*.

In analogy with the usual notation for the operation of a function **f**
on a real number **x**

we introduce the following notation for the operation of an operator on a function ,

The caret ``'' tells us that is a function operator which acts on the function immediately to its right, , giving the result .

Wed Oct 11 21:37:35 EDT 1995