I'm trying to do a simple simulation of a 1D charged quantum particle, which gets irradiated by an electromagnetic wave — in context of non-relativistic QM.

The Schrödinger equation for such a particle would be $$i\hbar\frac{\partial \Psi(x,t)}{\partial t}=\frac1{2m}\left(-i\hbar\frac{\partial}{\partial x}-qA(x,t)\right)^2\Psi(x,t)+q\left(\phi_0(x)+\phi(x,t)\right)\Psi(x,t),$$

where $A(x,t)$ is (what I believe to be) an $x$ component of vector potential $\vec A$ of radiation, $\phi_0(x)$ is static scalar potential of non-irradiated system, and $\phi(x,t)$ is scalar potential of radiation.

What I can't seem to understand is how electromagnetic radiation should be incorporated in this equation. I suppose it shouldn't depend on coordinate, i.e. in non-relativistic context speed of light should be considered infinite, thus wavelength should be infinite.

From basic knowledge of electromagnetism I suppose that such a wave would be represented by:

$$E=E_0 \cos(\omega t)$$ $$B=B_0 \cos(\omega t)$$

But simulation is to be 1D, while electromagnetic wave has three dimensions: one for direction of propagation (which I guess is irrelevant because of infinite wavelength), second for $\vec E$ and third one for $\vec B$.

To add the complexity, I need to determine how vector potential $\vec A$ and scalar potential $\phi$ would look in such case, which I might understand once I realize how to orient the radiation in terms of $\vec B$ and $\vec E$.

So, my question is: does it even make sense to talk about external electromagnetic radiation in 1D case? If yes, how should the field be oriented in this case to make a sensible quantum simulation? What would its vector potential $\vec A$ and scalar potential $\phi$ look like?