It is a misconception that one finds in the physical literature too often. The correct Coulomb energy is not static, but varies with time implicitly as

$$V_\mathrm{Coul} = V_\mathrm{Coul}(\boldsymbol{R}(t)) = \frac{ee}{4\pi\epsilon_0 |\boldsymbol{R}(t)|}$$

with $\boldsymbol{R}$ the interparticle distance. The problem is many people confounds this with the field-theoretic quantity

$$V_\mathrm{field} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x})\rho(\boldsymbol{y})}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|}$$

which is a static quantity, besides divergent (and unphysical). This confusion is not a problem exclusive to quantum theory, one already finds this kind of confusion in classical physics. In Ref. [1] you can find a detailed mathematical proof that the Coulomb potential $\phi(\boldsymbol{R}(t))$ is not reducible to the field-theoretic potentials $\phi(\boldsymbol{r},t)$.

If all of the above was not enough, there are additional misconceptions in quantum theory regarding the time dependence of operators. One finds too often the claim that operators in the Heisenberg picture vary with time, but they do not in the Schrödinger picture. This is not true, and in more advanced textbooks one finds expressions such as

$$i\hbar \frac{d}{dt} A_H(t) = [A_H(t), H_H(t)] + i\hbar \left( \frac{d}{dt} A_S(t)\right)_H$$

where the last term (missing in some textbooks) accounts for the time change of the operator in the Schödinger picture (subindex "S" is the operator in the Schrodinger picture and subindex "H" is in the Heisenberg picture). In what follow I work in the Schrödinger picture and avoid the subindex "S".

The point is that electrons and nuclei, all together attract or repel with non-electrostatic interactions, because electrons are never at rest in atoms.

In the nonrelativistic limit, the interaction between charges is given by the Coulomb potential $V_\mathrm{Coul}$. In the phase space formulation of quantum mechanics one simply uses this expression directly. In the Schrödinger wavefunction formulation one has first to replace position variables with Schrödinguer operators $\boldsymbol{x}\rightarrow \hat{\boldsymbol{x}}$; those operators maintain the implicit time dependence of the classical variables $\boldsymbol{x}=\boldsymbol{x}(t)$, and one obtains a Coulomb operator with a implicit time dependence $\hat{V}_\mathrm{Coul}(\hat{\boldsymbol{R}}(t))$.

This implicit time-dependence is not quoted in the literature, but it is needed to make sense of dynamical variables. For instance if we want to compute the velocity of an electron in the Schrödinger picture, we have first to obtain the velocity operator as

$$\hat{\boldsymbol{v}} = \frac{d\hat{\boldsymbol{x}}}{dt} = \frac{i}{\hbar} [\hat{H},\> \hat{\boldsymbol{x}}]$$

and then use this velocity operator in

$$\langle\boldsymbol{v}\rangle = \int d^3\boldsymbol{x} \>\Psi^{*}(t)\>\hat{\boldsymbol{v}} \>\Psi(t)$$

REFERENCE [1] https://journals.aps.org/pre/abstract/10.1103/PhysRevE.53.5373

It is a misconception that one finds in the physical literature too often. The correct Coulomb energy is not static, but varies with time implicitly as

$$V_\mathrm{Coul} = V_\mathrm{Coul}(\boldsymbol{R}(t)) = \frac{ee}{4\pi\epsilon_0 |\boldsymbol{R}(t)|}$$

with $\boldsymbol{R}$ the interparticle distance. The problem is many people confounds this with the field-theoretic quantity

$$V_\mathrm{field} = \frac{1}{2} \int \mathrm{d}^3 \boldsymbol{x} \int \mathrm{d}^3 \boldsymbol{y} \frac{\rho(\boldsymbol{x})\rho(\boldsymbol{y})}{4\pi\epsilon_0 |\boldsymbol{x}-\boldsymbol{y}|}$$

which is a static quantity, besides divergent (and unphysical). This confusion is not a problem exclusive to quantum theory, one already finds this kind of confusion in classical physics. In Ref. [1] you can find a detailed mathematical proof that the Coulomb potential $\phi(\boldsymbol{R}(t))$ is not reducible to the field-theoretic potentials $\phi(\boldsymbol{r},t)$.

If all of the above was not enough, there are additional misconceptions in quantum theory regarding the time dependence of operators. One finds too often the claim that operators in the Heisenberg picture vary with time, but they do not in the Schrödinger picture. This is not true, and in more advanced textbooks one finds expressions such as

$$i\hbar \frac{d}{dt} A_H(t) = [A_H(t), H_H(t)] + i\hbar \left( \frac{d}{dt} A_S(t)\right)_H$$

where the last term (missing in some textbooks) accounts for the time change of the operator in the Schödinger picture (subindex "S" is the operator in the Schrodinger picture and subindex "H" is in the Heisenberg picture). In what follow I work in the Schrödinger picture and avoid the subindex "S".

The point is that electrons and nuclei, all together attract or repel with non-electrostatic interactions, because electrons are never at rest in atoms.

In the nonrelativistic limit, the interaction between charges is given by the Coulomb potential $V_\mathrm{Coul}$. In the phase space formulation of quantum mechanics one simply uses this expression directly. In the Schrödinger wavefunction formulation one has first to replace position variables with Schrödinguer operators $\boldsymbol{x}\rightarrow \hat{\boldsymbol{x}}$; those operators maintain the implicit time dependence of the classical variables $\boldsymbol{x}=\boldsymbol{x}(t)$, and one obtains a Coulomb operator with a implicit time dependence $\hat{V}_\mathrm{Coul}(\hat{\boldsymbol{R}}(t))$.

This implicit time-dependence is not quoted in the literature, but it is needed to make sense of dynamical variables. For instance if we want to compute the velocity of an electron in the Schrödinger picture, we have first to obtain the velocity operator as

$$\hat{\boldsymbol{v}} = \frac{d\hat{\boldsymbol{x}}}{dt} = \frac{i}{\hbar} [\hat{H},\> \hat{\boldsymbol{x}}]$$

and then use this velocity operator in the ordinary Schrödinger-picture expression

$$\langle\boldsymbol{v}\rangle = \int d^3\boldsymbol{x} \>\Psi^{*}(t)\>\hat{\boldsymbol{v}} \>\Psi(t)$$

REFERENCE

[1] https://journals.aps.org/pre/abstract/10.1103/PhysRevE.53.5373