Time Development Example

Start off in the state.

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

In the Schrödinger picture,

\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}$x$\egroup.

\begin{eqnarray*}
\langle\psi\vert x\vert\psi\rangle &=&{1\over 2}\sqrt{\hbar\ov...
...ega t}\right) \\
&=&\sqrt{\hbar\over m\omega}\cos(\omega t)\\
\end{eqnarray*}


In the Heisenberg picture

\begin{displaymath}\bgroup\color{black}\langle \psi\vert x(t)\vert\psi\rangle={1...
...{-i\omega t} A
+e^{i\omega t}A^\dagger\vert\psi\rangle \egroup\end{displaymath}

This gives the same answer with about the same amount of work.



Jim Branson 2013-04-22