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.

$$\begin{eqnarray*} i\hbar{\partial\psi\over\partial t}=H\psi \\ \end{eqnarray*}$$

$$\psi(t)=e^{-iHt/\hbar}\psi(0)$$

where $H$ is the operator. We can expand this exponential to understand its meaning a bit.

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

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

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