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 (
A theoretical physics FAQ