Time Development Example

Start off in the state at \bgroup\color{black}$t=0$\egroup.

\begin{displaymath}\bgroup\color{black}\psi(t=0)={1\over\sqrt{2}}(u_1+u_2) \egroup\end{displaymath}

Now put in the simple time dependence of the energy eigenstates, \bgroup\color{black}$e^{-iEt/\hbar}$\egroup.

\begin{displaymath}\bgroup\color{black}\psi(t)={1\over\sqrt{2}}(u_1e^{-i{3\over ...
...sqrt{2}}e^{-i{3\over 2}\omega t}(u_1+e^{-i\omega t}u_2) \egroup\end{displaymath}

We can compute the expectation value of \bgroup\color{black}$p$\egroup.

\begin{eqnarray*}
\langle\psi\vert p\vert\psi\rangle &=&-i\sqrt{m\hbar\omega\ove...
...i\omega t}\right) \\
&=&-\sqrt{m\hbar\omega}\sin(\omega t) \\
\end{eqnarray*}




Jim Branson 2013-04-22