The Wavefunction for the HO Ground State

The equation

\begin{displaymath}\bgroup\color{black} Au_0=0 \egroup\end{displaymath}

can be used to find the ground state wavefunction. Write \bgroup\color{black}$A$\egroup in terms of \bgroup\color{black}$x$\egroup and \bgroup\color{black}$p$\egroup and try it.

\left(\sqrt{m\omega\over 2\hbar}x+i{p\over\sqrt{2m\hbar\omega}...
...dx}\right)u_0=0 \\
{du_0\over dx}=-{m\omega x\over\hbar}u_0 \\

This first order differential equation can be solved to get the wavefunction.

\begin{displaymath}\bgroup\color{black} u_0=Ce^{-m\omega x^2/2\hbar} .\egroup\end{displaymath}

We could continue with the raising operator to get excited states.

\begin{displaymath}\bgroup\color{black} \sqrt{1}u_1=\left(\sqrt{m\omega\over 2\hbar}x-\sqrt{\hbar\over 2m\omega}{d\over dx}\right)u_0 \egroup\end{displaymath}

Usually we will not need the actual wave functions for our calculations.

Jim Branson 2013-04-22