I have always had trouble connecting the picture of classical electromagnetism with the idea of photons. To make this connection better I'd like to ask the following question. How many photons, at an instant in time, are there inside a microwave while you are heating your food? Since microwaves are so large (~12cm) I figure it could either be one photon or many photons that together form the microwave.
[Physics] How many photons are in a microwave oven
electromagnetic-radiationmicrowavesphotonsquantum mechanics
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"Then comes quantum mechanics and says that light consists of particles of quantized energy, hf. Also, now these particles are modeled as probability waves obeying Schroedinger's equation, which gives the probability of observing a photon particle at some point in space at any given time."
Quantum theory of radiation does not work like that. In common formulation, there is no Schroedinger equation for "photon wavefunction"; the EM field is not described by multi-particle wave function $\psi(\mathbf r_1, \mathbf r_2, ...)$ of the kind one uses for electrons in an atom. Instead, the state of the EM field in a metallic cavity may be described by a ket vector in the Fock space $|\Psi\rangle$, which is a space of kets corresponding to a set of independent harmonic oscillators (tensor product space).
how does that change our model of the classical electromagnetic field? Do we now view it as some sort of average, or expectation value, of a huge number of individual photons emitted from a source? If so, how are the actual E⃗ and B⃗ values at a point (r⃗ ,t) calculated
It is a quantum theory of EM field, so it does not necessarily change the concept of the classical electromagnetic field in classical theory (the connection of the two theories is problematic). Within quantum theory, the properties of the classical electromagnetic field are best approximated by a special kind of Fock state, so-called coherent state. This state cannot be characterized as state with definite number of photons - the concept of photons is not well applicable to such states. The quantity resembling classical EM field is calculated from the Fock state as
$$ \langle \Psi | \hat{\mathbf E} |\Psi\rangle, $$
where $\hat{\mathbf E}$ is the operator of the electric field (this is an expression composed of the ladder operators of the harmonic oscillators and the vector eigenfunctions of the Helmholtz equation satisfying the boundary conditions for the cavity). In case the state $|\Psi\rangle$ is coherent, the above expression has similar mathematical properties to classical EM wave.
When dealing with a single quantum mechanical particle, both the wavefunction and the electric field appear to belong to the familiar class of "fields", both $\mathbf{E}(x)$ or $\psi(x)$. This analogy completely breaks down when you consider multiple particles, in which case the wavefunction depends on all of the particle coordinates, i.e. $\psi(x_1,x_2,x_3,\ldots,x_N)$. This is totally different from the behaviour of the "physical fields" such as the electromagnetic fields, which can be described by a function of a single coordinate, no matter how many particles one has in the system. These physical fields are included in our theories precisely because they enable us to describe physics in a local way. On the other hand, taking the object $\psi(x_1,x_2,x_3,\ldots,x_N)$ to be a physical field, which now propagates in a $3N$-dimensional configuration space, leads to a grossly non-local description that is philosophically abhorrent to many physicists.
Best Answer
Household microwave ovens operate at 2.45 GHz, apparently.
So the energy of a photons is $E=hf=6.63\times 10^{-34} \times 2.45 \times {10^9}=1.62 \times 10^{-24}$ Joules
So if you take an 800 Watt unit, that is putting $4.92\times 10^{26}$ photons per second into the microwave unit. Which is indeed `many photons'.
(That could be an overestimate by a factor of about 2, depending on whether your '800 Watt' microwave is one that consumes 800 W wall plug power, or one that puts a useful 800 W into the oven. I don't know the exact specification and a quick google search was unhelpful. )