Coherent States

A *coherent state*
is an eigenstate of the
annihilation operator with respect to the eigenvalue
:

An explicit formula for coherent states can be found by expanding
in terms of the eigenstates of the harmonic oscillator, or
*number states*
,
,
substituting into
the definition
(A.34) and
equating the coefficients of the
. The result is

and therefore . Equations (A.34) and (A.35) show that every is an eigenvalue of and thus defines a coherent state . Note that the are not pairwise orthogonal, as the scalar product is not zero for .

In order to come to a more intuitive understanding of coherent states
it is useful to consider WEYL's unitary
*displacement operator*
[MM71]

Formally, is obtained from the definition of the annihilation operator (A.24) by exchanging the operators , for the scalars , .

As indicated by its name,
the displacement operator
acts on a state by shifting
it in phase space. Using the BAKER-CAMPBELL-HAUSDORFF formula
(A.4) one can derive the product representation

such that in phase space

(A.29) |

Using
the displacement operator
one can now write a coherent state as

With this result it is now clear how the set of all coherent states
is to be interpreted:
it is obtained by
moving the ground state of the harmonic oscillator to all points of the
phase plane. Therefore the well-known properties of
,

In particular, the coherent states are *minimum uncertainty states*,
i.e. they are characterized by the smallest possible product of
standard deviations of the position and momentum operators
as given by the HEISENBERG uncertainty relation:

In the form (A.35) coherent states were first constructed by
SCHRÖDINGER, who
intended to use
them to demonstrate the ``continuous transition
from micro- to macromechanics'' [Sch26].
He showed that a wave packet (A.35), when specified as the
initial state submitted to the potential of the harmonic oscillator, does
not broaden during the
dynamics.^{A.8}With the according Hamiltonian

that is, again a (different) coherent state, except for a phase factor. Therefore the uncertainties , are conserved, which is just the expected behaviour for a classical particle.

The equations
(A.39-A.43)
also show that the state
of equation (A.29)
in fact is a coherent state up to a phase factor,

For this reason the antinormal-ordered distribution function is often called the

It is
interesting to note that the set of coherent
states
is
*overcomplete*:

Using the overcompleteness property (A.52) of the
coherent states and the expression (A.51) for
it is easy to show that the antinormal-ordered distribution
is normalized in the sense of

For more information on coherent states I refer the reader to the
specialist literature: the monograph [KS85]
remains
*the* standard work
in this field;
some more recent studies are
for example [WK93,KWZ94,ZK94],
where among other questions the problem of coherent states in
finite-dimensional HILBERT spaces is addressed.
Finally, in [Nie97a] NIETO presents an interesting
overview of the historic development of the theory of coherent states
as well as of the squeezed states which I
discuss in the following
subsection.

- ...
dynamics.
^{A.8} - But note that SCHRÖDINGER's generalizing interpretation in [Sch26] of this feature of the harmonic oscillator is erroneous. See [Str01b] for some background material on this issue.
- ...
particle.
^{A.9} -
Moreover, for the present case of the harmonic oscillator
EHRENFEST's theorem
allows to conclude
that the dynamics of the mean values
of the position and momentum operators coincide with the classical
dynamics of the
observables and , namely a harmonic
oscillation bounded by turning points which
are determined by the energy
:

with the phase shift .

- ... general.
^{A.10} -
By imposing certain additional conditions on the expansion
coefficients
it is still possible to achieve a unique expansion
in terms of
coherent states. More on this
*GLAUBER expansion*can be found in [Per93].