The Time Development Operator *

We can actually make an operator that does the time development of a wave function. We just make the simple exponential solution to the Schrödinger equation using operators.

i\hbar{\partial\psi\over\partial t}=H\psi \\

\bgroup\color{black}$\displaystyle \psi(t)=e^{-iHt/\hbar}\psi(0) $\egroup
where \bgroup\color{black}$H$\egroup is the operator. We can expand this exponential to understand its meaning a bit.

\begin{displaymath}\bgroup\color{black}e^{-iHt/\hbar}=\sum\limits_{n=0}^\infty{(-iHt/\hbar)^n\over n!}\egroup\end{displaymath}

This is an infinite series containing all powers of the Hamiltonian. In some cases, it can be easily computed.

\bgroup\color{black}$e^{-iHt/\hbar}$\egroup is the time development operator. It takes a state from time 0 to time \bgroup\color{black}$t$\egroup.

Jim Branson 2013-04-22