Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A1 / b

It represents the squared orbital angular momentum. In atomic units, $\hslash =1$, so its eigenvalues are $l(l+1)$. In ordinary units the corresponding observable is ${\hslash }^{2}l(l+1)$, with orbital angular momentum quantum number $l=0,1,2,\dots$.

The spherical harmonic is also an eigenfunction of the $z$ component of orbital angular momentum:These are atomic units; restoring units gives $m\hslash$. The integer $m$ is the magnetic quantum number.

Separation of variables makes the angular factor a spherical harmonic. Applying the Quantum Hamiltonian givesConsequently $R{Y}_{lm}$ is an energy eigenfunction provided the radial factor satisfiesThis is the radial Schrödinger equation for a central potential, here with the attractive Coulomb potential. An arbitrary $R$ does not give an eigenfunction; regularity, decay and wavefunction normalization select admissible bound quantum states.

For a $2p$ atomic orbital, $l=1$. The radial factor obeysSubstitution into the radial Schrödinger equation for a central potential givesThus the energy eigenvalue isThe normalization constant cancels; $E_{\mathrm{h}}$ is the Hartree energy.