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Look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$. $x$ is a point in configuration space and $t$ is the evolution parameter. They both look the same in the equation, then why consider one as an evolution parameter and other as configuration of the system.

My question is why should we even consider the evolution of the wave function in some parameter (it is usually time)?. Why can't we just deal with $\psi(\boldsymbol{x})$, where $\boldsymbol{x}$ is the configuration of the system and that $|\psi(\boldsymbol{x})|^2$ gives the probability of finding the system in the configuration $\boldsymbol{x}$?

One may say, "How to deal with systems that vary with time?", and the answer could be, "consider time also as a part of the configuration space". Why wonder why this could not be possible.