Is there a (semiclassical) electric field operator?

Question

So I come from a chemistry background, where the electronic structure of atoms and molecules is central. For practical purposes, we usually work with a charge density operator

$$ \hat{\rho}(r) = q \delta (r-r') \; ,$$

with $r'$ being the "particle coordinate" so that the expectation value is

$$ \langle \Psi|\hat{\rho}(r)|\Psi\rangle = \int \left| \Psi (r')\right|^2 q \delta (r-r') \;dr' = q \left| \Psi (r)\right|^2 \; ,$$

in accordance with the idea of the probability being proportional to the wavefunction squared.

Then, in some electrodynamics exercises (eg. in Jackson) we take this charge density, then solve Gauss's equation for the electric field, and conclude that the electric field of the quantum mechanical system is given by our result.

I'm wondering: is there some operator $\hat{E}(r)$ that would give us back this semiclassical electric field as $\langle \Psi|\hat{E}(r)|\Psi\rangle $?

Maybe I am using semiclassical wrong in this context, in my understanding if I get my charge density from quantum mechanics but solve the classical Maxwell equations, that can be called semiclassical. I am also aware that the electric field operator is defined in quantum electrodynamics as acting on the modes of the field, but I don't think I am looking for that operator (as I am not interested in field-matter interactions), nor can I understand QED derivations with a chemistry background sadly.

I've found this paper which introduced a polarization operator $\hat{P}(r)$ so that the polarization field will be the solution of Gauss's equation for the charge density (Eq. 3 of the paper), which seems to be similar to what I am looking for. I am wondering: is this polarization field the same as the electric field in this context? My understanding is that we can only get a well defined longitudinal field component by solving Gauss's equation, so it might very well be that the longitudinal component of this polarization field is the same as of the electric field that I am looking for.

So is there such an operator that would have the electric field of the charge density corresponding to our wavefunction as the expectation operator? And if yes, how is that different from the polarization field operator?

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EDIT for clarity, let's consider the example of a hydrogen atom.

The electronic wavefunction of the ground state of the hydrogen atom is (in the simplest units)

$$\Psi(r) = \frac{1}{\sqrt{\pi}}\exp{(-r)} \; .$$

I want to find the electric field of this electron. In order to do this, I'll first evaluate the classical charge density, then solve Gauss's law for the field.

The charge density - and here we employ the semiclassical idea - will be given as the expectation value of the charge density operator

$$ \rho(r) = \int \left| \Psi (r')\right|^2 q \delta (r-r') \;dr' = \frac{q}{\pi} \exp{(-2r)}\; .$$

To get the field from this classical charge density I just solve Gauss's law for the field

$$\nabla \cdot \vec{E} = \rho \; \rightarrow \; \vec{E}(r) = -\frac{q e^{-2 r} \left(2 r^2+2 r+1\right)}{4 \pi r^2 } \hat{e}_r \; .$$

This idea is clear to me.

My first question is:

1. Is there an operator directly for the electric field, so that instead of evaluating first the expectation value of $\hat{\rho}$ and solving Gauss's law I could directly evaluate $\hat{E}$?

While researching this question I've came across the concept of the polarization density. I am not familiar with polarization calculations at all, but to me it seems like there's an intimate connection between the electric field and the polarization field in this context. To be more exact, in this paper Eq. 29 is exactly what I find for the electric field for the hydrogen atom, however in the paper it is claimed that it is the polarization field of the atom.

My second question is:

2. Is this electric field operator that I am looking for exactly the same as the polarization field operator? If not, what's the difference?

Answer 1

As long as we talk about quantum mechanics, the electromagnetic field is usually taken to be non-quantized (unlike the electrons and nuclei of the atoms). That is, there is no need for introducing field operators and wave functions. The situation changes in quantum field theory, where one quantized the particles and the elecytromagnetic field on equal footing, and one can introduce the operators in the standard way, see Quantization of the electromagnetic field (and also this answer). However, this is largely beyond the scope of what seems to be required in the case of the OP.

Update: The particular ways of solving the Poisson equation for the electric potential, $$\nabla\cdot\mathbf{E}(\mathbf{r})=\nabla^2\phi(\mathbf{r})=\rho(\mathbf{r})$$ is by using Green's function, that is the solution of the equation for a point charge: $$\nabla^2G(\mathbf{r}, \mathbf{r}')=\delta(\mathbf{r}-\mathbf{r}'),$$ which gives us $$ G(\mathbf{r}, \mathbf{r}')=\frac{1}{4\pi}\frac{1}{|\mathbf{r}- \mathbf{r}'|}. $$ The potential from charge distribution $\rho(\mathbf{r})$ can now be written as $$ \phi(\mathbf{r})=\frac{1}{4\pi}\int d^3\mathbf{r}'\frac{\rho(\mathbf{r}')}{|\mathbf{r}- \mathbf{r}'|}.$$ One could also obtain a similar solution directly for the field, by defining a function $$\mathbf{F}(\mathbf{r},(\mathbf{r})')=\nabla_\mathbf{r}\cdot G(\mathbf{r}, \mathbf{r}')$$ (Working directly with the field is however less practical, since one needs to take care of all its components.)

See also this answer.