On the one hand, classical electromagnetism tells us that light is a propagating wave in the electromagnetic field, caused by accelerating charges. 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 Shrodinger's equation, which gives the probability of observing a photon particle at some point in space at any given time.
My question is – 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 $\vec{E}$ and $\vec{B}$ values at a point $(\vec r,t)$ calculated : how are they related-to/arise-from the probability amplitudes of observing individual photons at that point? Or put another way – how do the probability amplitude wavefunctions of the photons give rise to the electromagnetic vector field values we observe?
[In classical EM, if I oscillate a charge at frequency $f$, I create outwardly propagating light of that frequency. I'm trying to picture what the QM description of this situation would be – is my oscillation creating a large number of photons (how many?), with the $f$ somehow encoded in their wavefunctions?]
(Also, what was the answer to these questions before quantum field theory was developed?)
Best Answer
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).
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.