particle physics - What exactly is meant by the wavelength of a photon?

I've been thinking about this for quite some time, and from looking online I haven't found a satisfying answer.

Lots of photons, such as visible-light photons have very small wavelength (which from my understanding of basic physics is the distance between two crests/troughs), but I also know that some EM waves have wavelengths a few metres or even kilometres long e.g radio waves.

What keeps me up at night is the question "How can a photon have a wavelength of a few kilometres and yet still be thought of as a particle?"

Does this mean that one individual photon is several kilometres long? If so, wouldn't it be subject to so many variations between the beginning of the wave and its end?

I realise that matter is also wave-like, where it's uncertainty in position is given by its De Broglie wavelength. Does this apply to the photon?

In other words, is the wavelength of a photon simply the uncertainty in its position?

Answer

The photon is an elementary particle in the standard model of particle physics. It does not have a wavelength. It is characterized in the table as a point particle with mass zero and spin one. Its energy is given by E=h*nu, where nu is the frequency of the classical electromagnetic wave which can be built up by photons of the same energy.

This is where the confusion comes. The wavelength and frequency characterize the emergent electromagnetic wave from very many photons. How the classical wave emerges can be seen here although it needs a quantum field theory background to understand it. The photon, as a quantum mechanical entity, has a quantum mechanical wavefunction. This wavefunction complex conjugate squared gives the probability density for the specific photon to be at (x,y,z,t) . The frequency in the wavefunction is the frequency of the possible emergent classical wave, but for the individual photon it is only connected with probability of manifestation, as for example in the single photon double slit experiments.

single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames

you ask:

"How can a photon have a wavelength of a few kilometres and yet still be thought of as a particle?

It does not. It takes zillions of photons to build up the classical electromagnetic wave. In the photos above each individual photon gives a little dot. The build up gives the probability density distribution for photons, and lo, there is a frequency associated with the interference pattern, even though the photon manifests individually as a dot at the (x,y) of the screen.

That is why we need quantum mechanics.