Raising and Lowering Constants

We know that \bgroup\color{black}$A^\dagger$\egroup raises the energy of an eigenstate but we do not know what coefficient it produces in front of the new state.

\begin{displaymath}\bgroup\color{black} A^\dagger u_n=Cu_{n+1} \egroup\end{displaymath}

We can compute the coefficient using our operators.

\vert C\vert^2 &=&\langle A^\dagger u_n\vert A^\dagger u_n\ran...
...]) u_n\vert u_n\rangle= (n+1)\langle u_n\vert u_n\rangle=n+1 \\

The effect of the raising operator is

\begin{displaymath}\bgroup\color{black}A^\dagger u_n=\sqrt{n+1}u_{n+1}.\egroup\end{displaymath}

Similarly, the effect of the lowering operator is

\begin{displaymath}\bgroup\color{black} Au_n=\sqrt{n}u_{n-1} .\egroup\end{displaymath}

These are extremely important equations for any computation in the HO problem.

We can also write any energy eigenstate in terms of the ground state and the raising operator.

\begin{displaymath}\bgroup\color{black} u_n={1\over\sqrt{n!}}(A^\dagger)^nu_0 \egroup\end{displaymath}

Jim Branson 2013-04-22