Activity: Matrix Representation of Angular Momentum

Central Forces 2023 (2 years)

The operator \(\hat{L}_z\) that represents the \(z\)-component of angular momentum, the operator \(\hat{L}^2\) that represents the total angular momentum, and the operator \(\hat{H}\) that represents the energy for the rigid rotor (a particle confined to the unit sphere) have eigenvalues given by \begin{align} \hat{L}_z \left|{\ell, m}\right\rangle &=m\hbar \left|{\ell, m}\right\rangle \\ \hat{L}^2 \left|{\ell, m}\right\rangle &=\ell(\ell+1)\hbar^2 \left|{\ell, m}\right\rangle \\ \hat{H} \left|{\ell, m}\right\rangle &=\frac{\hbar^2}{2I}\, \ell(\ell+1)\left|{\ell, m}\right\rangle \end{align} Find the matrix representations for these operators.


Learning Outcomes