Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 33C (Principles of Quantum Mechanics)

The orbital angular momentum operator isUsing $[X_i,X_j]=[P_i,P_j]=0$ and $[X_i,P_j]=i\hslash {\delta }_{ij}$,andThus the rotation commutators for orbital angular momentum are

Contract the two Levi-Civita symbols:Moving momenta past positions with
$P_k{X}_{\ell }=X_{\ell }{P}_k-i\hslash {\delta }_{k\ell }$ gives the squared orbital angular momentum in position and momentum operators:Therefore

In position representation,andAlso $(r{\partial }_r{)}^2=r^2{\partial }_r^2+r{\partial }_r$. Substitution into the Cartesian identity givesbecause every radial derivative cancels. Hencethe spherical Laplacian from orbital angular momentum.

The first excited hydrogen eigenspace is the four-dimensional $n=2$ spaceThe first vector is the $2s$ state and the other three are the $2p$ states. All have the same unperturbed energy $E_2$.

The states $|n,\ell ,m⟩$ diagonalize $L^2$:The perturbation therefore introduces no mixing. The orbital-angular-momentum-squared perturbation of hydrogen leavesat energywhileremain triply degenerate atThese statements are exact because the unperturbed Hamiltonian commutes with $L^2$.

Part (a) givesHence both operators preserve the magnetic quantum number $m$. They are also odd under parity, while $|n,\ell ,m⟩$ has parity $(-1{)}^{\ell }$. Their matrix elements between states of equal parity vanish. The position-momentum selection rules in the first excited hydrogen eigenspace therefore leave only the coupling betweenThe $m=\pm 1$ states decouple, and all diagonal matrix elements vanish.

In the ordered basis $(|s⟩,|p⟩)$, remove the common energy $E_2$ and writeLetThe rotating-frame amplitudes obey a constant Schrödinger equationPutSince $H_{\mathrm{eff}}^2=q^{2}I$,For the initial amplitudes $(A_0,A_1)$, the exact coefficients are thereforeandRestoring the common unperturbed phase, the solution isThis is the rotating-frame solution for a harmonically driven two-level system.