The shape of photons and electrons

Thinking in intuitive pictures about quantum field theory phenomena soon reaches its limits; so one cannot make statements that hold under all circumstances. So let me describe some clearcut situations:

A free photon can have the shape of an arbitrary solution of Maxwell's
equation in vacuum. But only very special solutions are controllable
and hence useful for experiments or applications.

Upon production in a laser, photons are more or less localized (not
precisely, this is impossible, as photons cannot have an exact
position, due to the lack of a unique position operator with commuting
coordinates); often only in the transversal direction of the beam -
then you don't know where it is in the beam, except probabilistically.

For photons on demand (that you can program to transmit information)
you need to know when and where you transmit the photon, so it must
be well-localized.

Of course, a slit or a half-silvered mirror delocalizes a photon,
and only a measurement (or decoherence along the way) relocalizes it.
This enables interference effects.
In these cases, the photon stops being particle-like and behaves just
like an arbitrary excitation of the e/m field, i.e., like a wave.

The particle picture of light is good only in the approximation where
geometric optics is applicable. This has been known for almost 200
years now.

The paradoxes and the alleged queerness of quantum theory both have their origin in misguided attempts to insist on a particle picture where it cannot be justified.

For more details see the slides
Classical and quantum field aspects of light

Optical models for quantum mechanics

Whatever electrons ''are'', it is completely determined by their state
(wave function or density matrix). In an often quite meaningful sense,
electron's ''are'' the charge and matter distribution determined by
their state. For example, this is what atom microscopes ''see'' when
they look at matter, what chemist compute when they do quantum
molecular computations to predict a molecule's properties, and what
other matter responds to in the (often good) mean field approximation.

Like for photons, the shape of a single electron can have the shape
of (the squared modulus of) an arbitrary solution of the Dirac
equation in which only positive energies occur.

If an electron is prepared in a device, its positional uncertainty is
no bigger than the size of the relevant part of the preparing device.
Electrons in a typical electron beam are localized quite well
orthogonal to the beam direction, and are delocalized to some extent
in the direction of the beam, corresponding to the uncertainty in the
time when the electron was produced. In any case, this is very far
from a plane wave, which is uniformly distributed over 3-space.
Of course, later manipulations can delocalize the electron, but I
cannot imagine machinery for turning it into a plane wave state
extending over a large spatial region.
(Plane waves are primarily used in introductory quantum mechanics,
mainly for didactical reasons.)

Now suppose your electron beam has a very high speed so that the
uncertainty in the time where the electron is produced translates into
a spatial uncertainty of 1000km, and suppose also that the electron
can move 1000km without significant external interactions. Then it is
easy to imagine how its position is uniformly delocalized along the
beam and across the 1000km.

This results in a very long and thin cloud - we call that a (very low
intensity) beam.

See also: The Shape of Hadrons

Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ