Matrix Representations

Abstract operators become concrete once we fix a basis. In the $\{|\ell, m\rangle\}$ basis, the angular-momentum operators are finite matrices of size $(2\ell+1)\times(2\ell+1)$ . Writing them out makes the algebra tangible and connects directly to the gates of a quantum computer.

L_z is always diagonal

We label states by $\hat{L}_z$ , so $\hat{L}_z$ is automatically diagonal with the eigenvalues $\hbar m$ on the diagonal. For spin- $\tfrac12$ (basis $m = +\tfrac12, -\tfrac12$ ),

L_z = \frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \frac{\hbar}{2}\sigma_z.

For spin-1 (basis $m = +1, 0, -1$ ),

L_z = \hbar\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}.

Off-diagonal pieces from the ladder operators

The components $\hat{L}_x$ and $\hat{L}_y$ are not diagonal; we build them from the ladder operators whose matrix elements we already know,

\langle \ell, m' |\hat{L}_\pm| \ell, m\rangle = \hbar\sqrt{\ell(\ell+1) - m(m\pm 1)}\;\delta_{m',\,m\pm 1}.

For spin-1, $\hat{L}_+$ has the two entries $\sqrt{2}\,\hbar$ above the diagonal:

L_+ = \hbar\sqrt{2}\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \qquad L_- = L_+^\dagger.

Then $\hat{L}_x = \tfrac12(\hat{L}_+ + \hat{L}_-)$ and $\hat{L}_y = \tfrac{1}{2i}(\hat{L}_+ - \hat{L}_-)$ give the explicit $3\times3$ matrices.

Spin-1/2 and the Pauli matrices

For spin- $\tfrac12$ the construction reproduces the Pauli matrices exactly,

L_x = \frac{\hbar}{2}\sigma_x, \quad L_y = \frac{\hbar}{2}\sigma_y, \quad L_z = \frac{\hbar}{2}\sigma_z.

A useful check

One can verify the matrices satisfy $L_x^2 + L_y^2 + L_z^2 = \hbar^2\ell(\ell+1)\,\mathbb{1}$ . For spin-1 this gives $2\hbar^2$ times the $3\times3$ identity — a quick way to confirm you assembled the matrices correctly.

Try it

Construct the spin-1 raising operator $L_+$ as a $3\times3$ matrix in the basis ordered $m = +1, 0, -1$ , and return the matrix element $\langle 0|L_+|{-1}\rangle$ (with $\hbar = 1$ ). It should equal $\sqrt{2}$ .

Run your code to see the quantum state.