Harmonic Oscillator Raising Operator

We wish to find the matrix representing the 1D harmonic oscillator raising operator.

We use the raising operator equation for an energy eigenstate.

$$\bgroup\color{black}A^\dag u_n=\sqrt{n+1}u_{n+1}\egroup$$

Now simply compute the matrix element.

$$\bgroup\color{black}A^\dag _{ij}=\langle i\vert A^\dag \vert j\rangle=\sqrt{j+1}\delta_{i(j+1)}\egroup$$

Now this Kronecker delta puts us one off the diagonal. As we have it set up, i gives the row and j gives the column. Remember that in the Harmonic Oscillator we start counting at 0. For i=0, there is no allowed value of j so the first row is all 0. For i=1, j=0, so we have an entry for $A^\dag _{10}$ in the second row and first column. All he entries will be on a diagonal from that one.

$$\bgroup\color{black}A^\dag =\left(\matrix{ 0&0&0&0&...\cr ... ...r 0&0&0&\sqrt{4}&...\cr ...&...&...&...&... }\right)\egroup$$