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Decoupling of Equations in Quantum Mechanics

Recall that the time-dependent Schrödinger equation is

\begin{displaymath}i \hbar \frac{d \Psi({\bf r}, t)}{dt} = {\hat H} \Psi({\bf r}, t),
\end{displaymath} (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

\begin{displaymath}\Psi({\bf r}, t) = e^{- i {\hat H} t / \hbar} \Psi({\bf r}, 0).
\end{displaymath} (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}.
\end{displaymath} (31)


C. David Sherrill