In Section 3.2, we introduced the quadrature operator for a single-mode field (see (3.2.20)). Here we describe the properties of this operator and its eigenstates in more detail. We recall that the commutator of and is

(A4.1)
and that

(A4.2)
Further, the quadrature representation

of a single-mode state

is defined by the relation

. Here we show that these relations imply that the eigenstates of

cannot satisfy

*Kronecker*-delta orthogonality. For suppose that

has a countable set of orthogonal eigenstates

with eigenvalues

so that

(A4.3)
and

(A4.4)
The matrix element of the commutator on the left-hand side of (

A4.1) would then be

(A4.5)
whereas the matrix element of the right-hand side of (

A4.1) would be simply

. These clearly cannot be equal since the former is zero when

*m* =

*n*, whereas the latter is only non-zero when

*m* =

*n*. Hence the orthonormality in (

A4.4) is inconsistent with the commutator (

A4.1). The resolution of this difficulty is to employ delta

*function* orthogonality for the eigenstates

having a continuum of eigenvalues

, so that

(A4.6)
In doing this, we have extended the set of possible states beyond those of square-integrable functions which can be represented in Hilbert space. This is because

is not defined. Now taking matrix elements of (

A4.1) in the basis of eigenstates

we find

(A4.7)
(p.235)
Hence

(A4.8)
where we have used the form of the derivative of the delta function given in Appendix

2. This result together with the resolution of the identity in terms of the quadrature eigenstates, given in (

3.3.27), leads to a representation for the action of

since

(A4.9)
This is just the familiar position representation of the action of the momentum operator on the state

. The representation of the action of

on

is simply

since

(A4.10)
We can use (

1.3.33), (

1.3.36), (3.2.36) and (

A3.16) to write the quadrature eigenstate in the form

(A4.11)
From this, we can obtain the quadrature representation of any state. For example, the quadrature representation of the coherent state is

(A4.12)
in agreement with (

3.6.45), where

*k* is the final factor in (

A4.12).