quantum mechanics - Physical examples of wave function that is stationary in space and varying in time. Something like $Psi(x,t) = psi(t)e^{-ikx}$

We know time independent Schrodinger equation, where the wave function is stationary in time and varies in space. Simple examples are, particle in an infinite potential well, and the Hydrogen atom, which is a particle in a time independent, radially attractive electric field.

What I wonder and would like to know is that, is there any real world situation where the wave function is stationary in space and varying in time, something on the lines of "space-independent Schrodinger equation", if it exists. (I don't know its formula or its existence, even in theory).

Something like $$\Psi(x,t) = \psi(t)e^{-ikx}$$. I'd like to know if any simple examples exist and also its physical interpretations. I am just super curious. (The observable still being assumed as bounded self adjoint linear operators), everything about QM still intact.