How is the classical EM field modeled in quantum mechanics?

Question

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?)

Answer 1

"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.

Answer 2

You will find a good answer to this question in this blog posting by Motl . As you will see the answer is not simple to be described in two paragraphs.

My experimentalist's summary is that photons, the elementary particles that are exchanged where classical physics describes electromagnetic interactions, are also operated upon by quantum mechanical operators described by the potentials that appear in Maxwell's equations. This assigns to the individual particles constants connected with the potentials , in addition to their energy, which defines the frequency of the electromagnetic field through E=h*nu ( h Planck's constant).

Just like wave packets in mechanics of large bodies may easily have small Δx and Δp, the same fact works for wave packetals Ψ[A⃗ (x,y,z)]. In the classical ℏ→0 limit, you may extract classical electromagnetism i.e. Maxwell's equations from the corresponding quantum Heisenberg equations for the operators by simplifying the commutator and neglecting subleading terms in ℏ.

It goes on to describe another way of seeing the smooth continuity between quantum mechanical and classical electromagnetism, looking at the wave itself. The classical electromagnetic wave is composed by a huge number of photons which, in an ensemble build it up, as demonstrated in the blog memo.

Answer 3

This question seems to mix up real photons and virtual photons. Electric charge is a fundamental physical quantity and a property of matter. It constitutes a capacity to participate in the EM force. Charged particles accelerate one another (towards or away from each other) due to the exchange of virtual photons - these are not the same phenomenon as real photons, but they are related. EM waves correspond to real photons, which are emitted and absorbed by charged particles and, because massless, travel at c in a vacuum.

So to the question 'How do the probability amplitude wavefunctions of the photons give rise to the electromagnetic vector field values we observe?' the answer is: the EM vector field values are not given by the probability amplitude wavefunctions of the photons because they're not given by real photons at all but by charges and virtual photons which are not the same thing.