Decoupling of Equations in Quantum Mechanics

Recall that the time-dependent Schrödinger equation is

$$i \hbar \frac{d \Psi({\bf r}, t)}{dt} = {\hat H} \Psi({\bf r}, t),$$ (29)

where ${\bf r}$ represents the set of all Cartesian coordinates of all particles in the system. If we assume that ${\hat H}$ is time-independent, and if we pretend that ${\hat H}$ is just a number, than we can be confident that the solution is just

$$\Psi({\bf r}, t) = e^{- i {\hat H} t / \hbar} \Psi({\bf r}, 0).$$ (30)

In fact, this remains true even though ${\hat H}$ is of course an operator, not just a number. So, the propagator in quantum mechanics is

$${\hat G}(t) = e^{- i {\hat H} t / \hbar}.$$ (31)