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

Write $D_x=y{\partial }_z-z{\partial }_y$, $D_y=z{\partial }_x-x{\partial }_z$ and $D_z=x{\partial }_y-y{\partial }_x$. The product rule applied to a smooth test function $f$ givesThe mixed second partial derivatives cancel because they commute. Since the orbital angular momentum components are ${\hat{j}}_i=-i\hslash D_i$, their commutator isThis is one of the orbital angular momentum commutation relations; the Planck constant enters through the definition of each linear operator.

Cyclic permutation of the coordinates in the preceding commutator calculation gives the remaining orbital angular momentum commutation relations:Reversing the order of a commutator reverses its sign.

Use the commutator derivation identity $[A,BC]=[A,B]C+B[A,C]$ and the orbital angular momentum commutation relations. They giveThe first two expressions cancel on addition, soThe same argument works for each component of the angular momentum operator.

The squared angular momentum operator and one selected component are commuting operators, so a complete set of common eigenfunctions can be chosen: for the $z$ component these are the usual $|j,m⟩$ angular momentum eigenstates. Distinct components do not have a common complete eigenbasis, because their commutator is another nonzero component. A $j=0$ state is an exception in which all components vanish; noncommutation does not forbid every individual common eigenfunction.

The squared angular momentum and one chosen component can have simultaneously sharp values. All three components cannot have simultaneously sharp values in a nonzero-$j$ state. The possible measured component values are $m\hslash$, while the squared magnitude is $j(j+1){\hslash }^2$. This concerns the quantum state, rather than imperfect experimental resolution. The Robertson uncertainty relation, for example,quantifies the restriction; its right-hand side may vanish in some states, without making all three components simultaneously sharp.

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.