The representation of the momentum operator
The first step is to imagine the form of the translation operator. Here we simply we simply consider a 4 components out of an n-dimensional space. We examine how the components xi’s are related to the translated components yi.
If one steps on step forward by one step (Choose the step to be x2-x1 so that the old component of basis e1 is now the component of basis e2.)
This is a shift of the function over one step. Similarly if
you go to the continuum and consider the step 
The only contribution is for the state exactly a distance away.
Now for small
(don’t worry
about i or the sign)
Where we are looking to be able to express the translation
in terms of some property of the current state times the amount or step for the
translation. This separates 
and its properties from the amount of the translation. Operators that have this characteristic are
referred to as generators. They
determine the amount-of-change/translational-step required.
In general, one can write an operator in a format similar to the matrix formulation used for finite dimensional vector spaces.
The use of the unit vectors in the matrix representation of
an operator is not always used. Notice the matrix formalism requires the
determination of all the 
’s and to use these elements to find the ith new component by
looking at all of the contributions from each part of the original vector. The particular
contribution of the jth component to the ith is . Detailing which
parts of the old and new vector are included is the same as projecting out the
contributions with an inner product. Thus the matrix representation could also
be written as
Now we are looking for a way to separate the amount from
some generic features of the operator.
From the intuitive standpoint a rotation involves a certain type of
operation performed over a given angle.
Rotation can be separated from the angle. One knows what a rotation is without
specifying how much rotation is involved.
Can one mathematically separate operations in the same fashion? For operations that are continuously
connected to the identity one can try and use a carries the essence of the translation and the parameter provides the
amount. For very small steps the
translation becomes
For large steps the translation takes the form
(Factors of i
and 
have been dropped.)
If you want to write a translation in terms of some generic
result times the step then the operator cannot depend on the step size. Clearly
to first order you could imagine that the slope at that point times the change would give you the
total change at that point. The first term 1 just involves keeping the
amplitude for each state the same. The second term 
finds the change per
step time the step size. Examine the effect on the value y2 based on the
original vector x. We need to find the change in the amplitudes and we base it
on the change to the right averaged with the change to the left. There might be
Where x1 has been replaced by the amplitude 
evaluated at x1. So we
have constructed our space out of discrete states each one a distance 
from the previous one. The matrix then grabs the amount of
three adjacent basis states and combines them to produce a derivative like
expression. This matrix has the form as shown above
This has the correct behavior wrt the sign. The transpose of this matrix is minus the original.
In principle one can imagine a matrix that links the adjacent states in such a way as to calculate the change and then use the change as the first approximation in computing the new vector by adding the change to the current vector. It is an amazing feature of function theory that the value of a function can be computed at an arbitrary distance away if the function and all its derivatives at some point are known. This feature allows us to take rotations, time evolution and special translation and express them in terms of generators and the amount.
This discussion was meant to review some aspects of the
definition of the momentum operator in terms of the derivative 
. This note is meant
to help clarify how the form of this operator emerges as a derivative.