Why do we assume the potential is independent of time in the Schrödinger equation?

Question

In just about every text I read (online or in paper), when they handle the time-dependent Schrödinger Equation, I see something along the lines of "we always assume the potential is independent of time." Why is this? Are there not plenty of circumstances when this isn't valid? Aren't most experiments done with varying potentials (NMR for example, the magnetic field, which affects the potential, is changing in time)? Is this assumption made in textbooks just for pedagogical reasons, to make life easier?

If we don't make this assumption, then it seems to me that the Schrödinger equation is no longer separable and we can no longer just apply the time-evolution operator as is usually done (and the time-independent equation is no longer valid).

Perhaps tangential to the main question but: Also, if we want to solve it numerically, it seems to me we also can't simplify using split-step Fourier transforms or into a form handled by Runge-Kutta. Is this correct? I'm especially interested in exploring the numerical analysis, but I guess I should post that question in the scientific computing SE.

Of course, when I say "potential" I mean $V\left(\vec r, t\right)$ in the equation \begin{equation} i\hbar\frac{\partial}{\partial t} \Psi\left(\vec r, t\right) = \left[\frac{-\hbar^2}{2m}\nabla^2+V(\vec r, t)\right]\Psi\left(\vec r, t\right) \end{equation} and the assumption whose justification I don't understand is $V\left(\vec r, t\right)=V\left(\vec r\right)$.

Answer 1

There are plenty of situations where the potential depends on time. The core reason you haven't seen them is likely that you haven't been looking in the right places.

However, that said, there is indeed a clear separation between the static and the time-dependent components of the potential. For the vast majority of experiments where we use a time-dependent probe to interact with the system, the probe is extremely weak (by several orders of magnitude) when compared to the natural hamiltonian of the system. This means that it is best treated using perturbation theory, so that the best strategy is to solve the time-independent Schrödinger equation for the dominating structural part of the hamiltonian (which generally doesn't depend on time) and then worry about the probe.

Moreover, a huge number of experiments are done, for various reasons, using oscillating potentials which are very close to monochromatic. For those potentials, it is often possible to move to a 'rotating frame' in which the interaction hamiltonian effectively becomes static, which makes the analysis much simpler.

Still, there's plenty of situations where none of this is valid, particularly if the probe is strong enough to get out of the perturbative regime. But even then, it is still important to have the structure of the system (i.e. the eigenstates of the interaction-free hamiltonian) at hand, as they are generally important parts of the analysis, even when they no longer play an explicit role in solving the TDSE.

If you want a deeper exploration of these themes, I recommend David Tannor's Quantum Mechanics: A Time-Dependent Perspective.

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And finally,

Also, if we want to solve it numerically, it seems to me we also can't simplify using split-step Fourier transforms or into a form handled by Runge-Kutta. Is this correct?

No, it's not. Time-dependent potentials are perfectly solvable using the standard numerical methods. They might need a small bit of fine-tuning, but nothing more.