What is the physical meaning of a single photon's frequency?
When the fields cross zero, is there less probability of finding an photon there? What does the amplitude of the waves represent, if anything?
And if the frequency has to do with probability, how to interpret circularly polarized light where there's no point that the magnetic and electric fields are both zero?
"What is the physical meaning of a single photon's frequency?" "What does the amplitude of the waves represent, if anything?"
Let's first start by asking what a single photon is, and how it connects to macroscopic light. A single photon is typically in layman's terms described as the smallest bit that an electromagnetic wave can exist as. This understanding doesn't exactly illuminate some of the properties of what a photon really is, so let's talk a little bit more clearly what we mean by a photon.
Typically, the way that you introduce the notion of a photon is first by starting by taking classical waves and applying quantum mechanical rules to them.
Classical E&M waves can be obtained by solving Maxwell's Equations: $\sum_k A_k Cos(kx-w(k)t)$.
We can write any propagating wave as a superposition of these individual frequencies. We're going to take a single one of these frequencies and apply quantum rules to it. In doing so we will quantize light for a particular frequency mode (and we will be able to talk about what a single photon would be for that particular mode)
Now we know that the energy of the electric field is $E^2 +c^2B^2$. This equation has the same form as the energy of a harmonic oscillator $p^2+wx^2$, so if we treat E & B as X & P, then we see that we can use the solutions we have for a particle in a harmonic oscillator as the solutions for the E & B field! As we know from a particle in a harmonic oscillator potential, there is a groundstate energy (and wavefunction) that's nonzero. This is the energy in the vacuum and describes the gaussian-shaped uncertainty in the E-field even with zero photons. Now the next highest energy state is the "single-photon" also know as "the first Fock state." This has a wavefunction associated with it that has a probability distribution of measuring the E-field at particular values.
So finally, to summarize, a photon's frequency is actually very well defined. This is because frequency is in the same units as energy, and we've identified that there are distinct energies allowed for a particular frequency-mode. The interesting quantum effects of being a single photon have to do with the amplitude of the E-field and not its frequency (or energy).
When the fields cross zero, is there less probability of finding an photon there?
The average E-field amplitude value of any fock-state is actually zero! The standard deviation of any photon-number state is what changes with increased photon number. To get non-zero average photon number you need to have a quantum superposition of photon number states!
Also measuring a zero-Efield does not tell you that much about what your state is. You need to do repeated identical measurements in order to indentify what your initial state is (to reconstruct the wavefunction distribution)
