quantum mechanics - Who is doing the normalization of wave function in the time evolution of wave function?

In the Schrödinger equation, at any given time $t$ we should jointly add another sub equation, like

$$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations simultaneously. Why not? I know it does not yield, but I am always baffled, who is doing the normalization of the wave function? Observers, the system, the measuring process, God?