Harmonic Oscillator Lowering Operator

We wish to find the matrix representing the 1D harmonic oscillator lowering operator. This is similar to the last section.

The lowering operator equation is.

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

Now we compute the matrix element from the definition.

\begin{displaymath}\bgroup\color{black}A_{ij}=\langle i\vert A\vert j\rangle=\sqrt{j}\delta_{i(j-1)}\egroup\end{displaymath}


\begin{displaymath}\bgroup\color{black}A=\left(\matrix{
0&\sqrt{1}&0&0&0&...\c...
...0&0&0&\sqrt{4}&...\cr
...&...&...&...&...&... }\right)\egroup\end{displaymath}

This should be the Hermitian conjugate of \bgroup\color{black}$A^\dag $\egroup.



Jim Branson 2013-04-22