Consider the vector space L²_x of square integrable functions.  We have earlier identified two orthonormal, continuously labeled bases for L²_x,

 ,

and

.

We can generalize to three dimensions. Then the bases are

,

and

.

Let us associate the kets |p₀> with and |r₀> with , and the ket |y> with the wave function y(r).
Then denotes the components of |y> in the {|p₀>} basis and denotes the components of |y> in the {|r₀>} basis.

Recall:

For the {|p>} basis we have ,

and for the {|r>} basis we have .

The scalar product of two vectors |y> and |f> can be calculated on the{ |r>} basis and the {|p>} basis.

and

.

We use the transformation matrix U to change from the {|r>} basis to the {|p>} basis.

  .

For any change of basis with U we have shown

,

where d_k denotes the coordinates in the new basis and c_i denotes the coordinates in the old basis.

Therefore

.

  denotes the coordinates of |y> in the {|p>} basis and denotes the coordinates of |y> in the {|r>} basis.  If we denote the matrix elements of the operator A, <r|A|r’>, by A(r,r’) and the matrix elements <p|A|p’> by A(p,p’) then

 ,

which corresponds to

.

The R and P operators

Let X be the operator whose eigenvectors are {|r>}.
Let X|r>=x|r>, with x a real number. 
Let P_x be the operator whose eigenvectors are {|p>}. 
Let P_x|p>=p_x|p>, with p_x a real number. 
Make similar definitions for Y an Z and for P_y and P_z.  Let |y> and |f> be arbitrary kets.
Calculate

.

X is Hermitian.  In the {|r>} representation operating with the operator X just amounts to multiplying by the number x.

Similarly:

.

P_x is Hermitian.  In the {|p>} representation operating with the operator P_x just amounts to multiplying by the number p_x.

But we may also write

,

since P_x is Hermitian.

 is the Fourier transform of y(r).

But what is p_xy(p)?

The Fourier transform of is

 .

,

using

We therefore have

.

Therefore

since .  We have

.

In the {|r>} representation the operator P_x coincides with the differential operator .

Similarly, .

In the {|p>} representation the operator X coincides with the differential operator , etc.

The commutator [X,P_x]

.

=

for any and any |r>.  Therefore .  We proceed similarly for the y and z components.  We write

.

The operators R_i and P_i do not commute.  A common eigenbasis of R_i and P_i does not exist.