Cyberkatru wrote:
Thus either we (a) box-normalise, and say there is one

particle per box (of sides L), and hence have some sort of beam;
or we (b) restrict the domain (e.g. to between -L/2 to +L/2) so the
the integral is finite, and hence normalizeable (although then
exp(ikx) is unlikely to be a good wavefn).

--


I am sorry to say that this just doesn't make sense to me. I sounds like
hand waving. One particle per box? Again, if there a several fermions then
there sould be many variables in the wave function- an xyz triple for every
particle. That what the "official formalism" states.

It sounds like hand-waving but has a completely rigorous formal basis.
The claim was about Bosons, not fermions.

The physical state described by a typical laser beam is a state with
an indeterminate number of photons, since it is not an eigenstate of
the number operator. (This essentially means that a certain number
of photons cannot be meaningfully asserted.)
Thus the traditional N-particle picture does not apply.

Instead one has to work in a suitable Fock space. There, each mode
function (solution of the free Maxwell equation) A(x) gives rise to
a coherent state ||A in the Maxwell Fock space, an associated
annihilation operator a(A) = integral A(x) a(x) dx, and the
corresponding creation operator a^*(A) = a(A)^*.

These produce a single-mode Fock subspace consisting of all |A,psi,
where psi is the unnormalized wave function of a harmonic oscillator;
|psi|^2 is the intensity of the beam.
The coherent state itself corresponds to the normalized vacuum state
of the harmonic oscillator, ||A = |A,vac.

The Maxwell Fock space is the closure of the space spanned by all
the ||A,psi together. [It is not a direct product, though.]
This space is the pure electromagnetic field sector of QED,
describing a physical vacuum, i.e., a region of the universe
where matter is absent.

In optics experiments, laser beams are often idealized by ignoring their
extension perpendicular to the transmission direction. Then each beam
can be described by some ||A,psi. A coherent pair of laser
beams obtained by splitting is described by a superposition
||A_1,psi_1 + ||A_2,psi_2 of the two beams.

Beams of thermal light (such as that from the sun) and pairs of
beams created by independent sources, cannot be described by wave
functions alone, but need a density formulation. A single light beam
is then described (in the same idealization) by a mode A and a density
matrix rho in a single-mode Fock space, while k light beams are
described by k modes A and a density matrix rho in a k-mode Fock space.

In many treatments, the modes are left implicit, so that one works
only in the k-mode Fock space. This simplifies the presentation, but
hides the connection to the more fundamental QED picture.
For a thorough study of the latter, see the bible on quantum optics,
L. Mandel and E. Wolf,
Optical Coherence and Quantum Optics,
Cambridge University Press, 1995.


Arnold Neumaier