Electric fields and de broglie wavelength

[ascky]

I'm confused as to how the concept that 'things' have a wavelength applies to an electromagnetic field, which has momentum, but not mass: what 'things' have wavelengths? Are there other criteria, other than whether something has mass or momentum, that must be fufilled before talking about something having a wavelength?

 

[vanesch]

Are there other criteria, other than whether something has mass or momentum, that must be fufilled before talking about something having a wavelength?

Momentum is sufficient.

 

[ascky]

So how would one go about calculating the wavelength of, say, a dc electric field?

 

[vanesch]

So how would one go about calculating the wavelength of, say, a dc electric field?

The description in terms of de Broglie wavelength is - you might imagine - rather crude. In reality, you have to solve a mathematical problem to find an entire wave function, and the concept of wavelength only applies to very special kinds of solutions, namely sine waves. This is not going to be always possible (in fact, it is only possible in empty space, when the things are propagating freely, and even then not all solutions have to take on that aspect).
Your DC field is such an example: the DC field will be satisfying boundary conditions (conductors at certain potentials, for instance) which will make things such that a sine wave is not going to be possible as a solution.
If you want to put a DC field in empty space, you have only one option: a constant field. Such a field has of course a wavelength equal to infinity. To all other DC fields, with boundaries, the concept of "wavelength" itself becomes inappropriate because the solution is not a sine function.