Harmonic Oscillator Hamiltonian Matrix

The basis is the harmonic oscillator energy eigenstates. We know the eigenvalues of $H$.

$$\bgroup\color{black}Hu_j=E_ju_j\egroup$$

$$\bgroup\color{black}\langle i\vert H\vert j\rangle=E_j\delta_{ij}=\left(j+{1\over2}\right)\hbar\omega\delta_{ij}\egroup$$

The Kronecker delta gives us a diagonal matrix.

$$\bgroup\color{black}H=\hbar\omega\left(\matrix{ {1\over 2}&0... ...0&0&0&{7\over 2}&...\cr ... &... &...&... &... }\right)\egroup$$