The |r> and |p> representation

Consider the vector space L²_x of square integrable functions. We have earlier identified two orthonormal, continuously labeled bases for L²_x,
{v_p(x) = (2πħ)^-1/2 exp(ipx/ħ)},
with Ψ(x) = ∫ Ψ(p) v_p(x) dp = (2πħ)^-1/2∫_-∞^∞ Ψ(p) exp(ipx/ħ) dp,
and
{δ_xo = δ(x - x₀)} with Ψ(x) = ∫_-∞^∞ Ψ(x₀) δ(x - x₀) dx₀.
We can generalize to three dimensions. Then the bases are
{v_p(r) = (2πħ)^-3/2 exp(ip∙r/ħ)}
with Ψ(r) = ∫ Ψ(p) v_p(r) d³p = (2πħ)^-3/2∫_-∞^∞ Ψ(p) exp(ip∙r/ħ) d³p,
and
{δ_ro = δ(r - r₀)} with Ψ(r) = ∫_-∞^∞ Ψ(r₀) δ(r - r₀) dx₀.

Let us associate the kets |p₀> with v_p0(r) and |r₀> with δ_ro, and the ket |Ψ> with the wave function Ψ(r).
Then Ψ(p) = <p₀|Ψ> denotes the components of |Ψ> in the {|p₀>} basis and Ψ(r) = <r₀|Ψ> denotes the components of |Ψ> in the {|r₀>} basis.
Recall:
For the {|p>} basis we have <p|p'> = δ(p - p'), ∫ d³p |p><p| = I,
and for the {|r>} basis we have <r|r'> = δ(r - r'), ∫ d³r |r><r| = I.

The scalar product of two vectors |Ψ> and |Φ> can be calculated in the {|r>} basis and the {|p>} basis.
<Φ|Ψ> = ∫ d³r <Φ|r><r|Ψ> = ∫ d³r Φ*(r)Ψ(r)
and
<Φ|Ψ> = ∫ d³p <Φ|p><p|Ψ> = ∫ d³p Φ*(p)Ψ(p)
We use the transformation matrix U to change from the {|r>} basis to the {|p>} basis.
U_r,p = <r|p> = v_p(r) = (2πħ)^-3/2 exp(ip∙r/ħ).
For any change of basis with U we have shown d_i = ∑_jU^†_ija_j,
where d_i denotes the coordinates in the new basis and a_j denotes the coordinates in the old basis.
Therefore
Ψ(p) = ∫d³r (2πħ)^-3/2 exp(-ip∙r/ħ) Ψ(r) = ∫ d³r U_r,p* Ψ(r).
Ψ(p) denotes the coordinates of |Ψ> in the {|p>} basis and Ψ(r) denotes the coordinates of |Ψ> 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
A(p,p’) = (2πħ)^-3∫d³r∫d³r' exp(-i(p - p')∙r/ħ)A(r,r’),
which corresponds to Ω_i'j' (new basis) = (U^†ΩU)_ij (old basis).

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 |Ψ> and |Φ> be arbitrary kets.
Calculate
<Φ|X|Ψ> = ∫ d³r x<Φ|r><r|Ψ> = ∫ d³r Φ*(r) x Ψ(r) = <Ψ|X|Φ>*.
X is Hermitian. In the {|r>} representation operating with the operator X just amounts to multiplying by the number x.
Similarly:
<Φ|P_x|Ψ> = ∫ d³p Φ*(p) p_x Ψ(p) = <Ψ|P_x|Φ>*.
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
<Φ|P_x|Ψ> = ∫ d³r <Φ|r><r|P_x|Ψ> = ∫ d³r Φ*(r) ∫ d³p <r|p><p|P_x|Ψ>.
But ∫ d³p <r|p><p|P_x|Ψ> = (2πħ)^-3/2∫p_x Ψ(p) exp(ip∙r/ħ) d³p,
since P_x is Hermitian.
Ψ(p) is the Fourier transform of Ψ(r).
Ψ(p) = (2πħ)^-3/2∫Ψ(r) exp(-ip∙r/ħ) d³r,

But what is p_xΨ(p)?
The Fourier transform of ∂Ψ(r)/∂x is
(2πħ)^-3/2∫_-∞^+∞∂Ψ(r)/∂x exp(-ip∙r/ħ) d³r
= (2πħ)^-3/2∫∫dy dz exp(-i(p_yy + p_zz)/ħ )∫dx exp(-ip_xx/ħ)∂Ψ(x,y,z)/∂x.

∫_-∞^+∞dx exp(-ip_xx/ħ)∂Ψ(x,y,z)/∂x
= exp(-ip_xx/ħ) Ψ(x,y,z)|_-∞^+∞ - ∫dx (-ip_x/ħ) exp(-ip_xx/ħ) Ψ(x,y,z),
using ∫dx u (dv/dx) = uv - ∫dx (du/dx)v.
exp(-ip_xx/ħ) Ψ(x,y,z)|_-∞^+∞ = 0.
We therefore have
(2πħ)^-3/2∫(ħ/i)∂Ψ(r)/∂x exp(-ip∙r/ħ) d³r
= p_x(2πħ)^-3/2∫Ψ(r) exp(-ip∙r/ħ) d³r = p_xΨ(p).
Therefore <r|P_x|Ψ> = ∫ d³p<r|p><p|P_x|Ψ>
= (2πħ)^-3/2∫p_x Ψ(p) exp(ip∙r/ħ) d³p
= (2πħ)^-3∫d³r'∫d³p exp(ip∙(r - r')/ħ) (ħ/i)∂Ψ(r')/∂x'
= (ħ/i)∂Ψ(r)/∂x,
since (2πħ)^-3∫d³p exp(ip∙(r - r')/ħ) = δ(r - r').
We have <r|P_x|Ψ> = (ħ/i)∂Ψ(r)/∂x.

In the {|r>} representation the operator P_x coincides with the differential operator (ħ/i)∂/∂x.
Similarly, P_y --> (ħ/i)∂/∂y, P_z --> (ħ/i)∂/∂z, P --> (ħ/i)∇.
In the {|p>} representation the operator X coincides with the differential operator (-ħ/i)∂/∂p_x, etc.

The commutator [X,P_x]
<r|X|ζ> = x<r|ζ> for any |ζ>, <r|P_x|ζ> = ((ħ/i)∂/∂x)<r|ζ> for any |ζ>.
<r|[X,P_x]|ζ> = <r|XP_x|ζ> - <r|P_xX|ζ> = x <r|P_x|ζ> - ((ħ/i)∂/∂x)<r|X|ζ>
= x ((ħ/i)∂/∂x)<r|ζ> - ((ħ/i)∂/∂x)(x<r|ζ>) = -(ħ/i)<r|ζ>,
for any |ζ> and any |r>. Therefore [X,P_x] = iħ I. We proceed similarly for the y and z components. We write
[R_i,P_j] = iħδ_ij, [R_i,R_j] = [P_i,P_j] = 0. (i, j = 1, 2, 3)
The operators R_i and P_i do not commute. A common eigenbasis of R_i and P_i does not exist.