I do understand that the solution obtained from the separation of variables to a system where $V$ is constant, is in a stationary state. Mathematically, I do get that the probability density only depends on $x$ since the time term of the wave function cancels out to one.
But, conceptually, how is it possible that after a long time, when the wave function has ceased, the probability density to not change? I've always thought that the probability density was a statistical representation of changing quantum states given by the wave function. So, it is confusing to think how the probability density exists same as before when the wave equation bare does.