This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.
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.