There are three simplest forms for function operators which we shall see over and over again in this course, the constant operator, the argument multiplication operator and the differential operator. These are described in detail in the three paragraphs below.
One of the simplest function operators with which we shall deal is
just multiplication of the input function by some constant, c. We
would then write this operator as 
so that the action of
is just
Note that, with the hat ``on,'' 
is an operator whereas with the
hat ``off'' c is just a constant number which is multiplying
. Two very
importation constant operators are the zero operator 
, whose
operation on any function returns the zero function, 
, and the unit operator 
whose action on any function is to
``do nothing'' or just return the input function back as output, 
.
We saw an example of our next more complicated operator in our first
expression for 
Here 
is derived from 
by multiplication of 
by its
argument x. For this new function operator, we write 
so that the action of 
on 
is just
We now may rewrite our expression for the average
position as 
The same type of operation
occurs in the calculation of the average momentum within the
momentum representation. In this case we define 
so that
and
. Note that we put the tilde
``
'' on the p below the hat to remind us that this
operator is defined for use in the momentum representation.
We also had an expression for the average position computed within the
momentum representation,
. Here, 
is derived from 
using the yet
more complicated operation of differentiation. We now write 
as a
differential operator 
so that the action
of 
is just
and the average position computed in the momentum representation is
. (Again, we put the tilde ``
'' on the x to remind us that this operator is defined for use in
the momentum representation.) Finally, to complete the set of
average-operator correspondences, for computing the average momentum
from within the position representation we define a second
differential operator 
so that
and
