Past exam of the advanced quantum physics course of the University of Cambridge / 2019 / Section A

The closed $1s^2$ and $2s^2$ subshells contribute zero total orbital angular momentum and spin, so only the two $2p$ electrons matter. The first of Hund's rules maximizes their total spin, giving $S=1$. The Pauli exclusion principle then requires an antisymmetric spatial wavefunction. Two parallel-spin electrons cannot occupy the same $m_l$ orbital: the greatest possible orbital projection is $1+0=1$, so the maximal allowed $L$ within this triplet is $L=1$.

The ground term is therefore ${}^{3}P$, in term symbol notation ${}^{2S+1}{L}_J$. Since the $p^2$ subshell contains fewer than half of its six possible electrons, the third of Hund's rules places the smallest $J$ lowest. Addition of angular momentum gives $J=0,1,2$, in that energy order within the ground term. ThusThese are the fine structure partners of the same electronic term, with magnetic multiplicities $1,3,5$, rather than different principal-shell excitations. The ordering agrees with the tabulated lowest neutral-carbon levels.^[ref]

Let $A_{21}$ be the spontaneous emission coefficient, and let $B_{12}\rho$ and $B_{21}\rho$ be the absorption and stimulated emission rates per atom under the radiation spectral energy density $\rho$ at resonance. Because the two levels are nondegenerate, the Einstein coefficients satisfy $B_{12}=B_{21}=B$.

The radiative balance in a nondegenerate two-level atom isThe total spontaneous and stimulated emission rates are $R_{\mathrm{sp}}=N_2{A}_{21}$ and $R_{\mathrm{st}}=N_{2}B\rho$. Dividing the balance equation by $N_{2}B\rho$ givesThis presumes nonzero drive and upper-level population, and no other population-transfer mechanism. It does not require a thermal Boltzmann distribution. In this closed driven two-level system, spontaneous decay keeps $N_1>N_2$; arbitrarily intense driving approaches equal populations, rather than producing a population inversion.

Use the electronic weak-field Zeeman splitting model, neglecting hyperfine structure. The Landé g-factor is $g_g=2$ for $3S_{1/2}$ and $g_e=2/3$ for $3P_{1/2}$. The shift of an absorption frequency is thereforeHere the magnetic field is weak relative to spin-orbit coupling, so $J$ remains a useful quantum number, but the spectral line widths are small enough to resolve its magnetic substates.

For linear polarization parallel to $z$, the optical electric field has only the field-axis $\pi$ component. The electric-dipole selection rule is $\Delta m=0$. Its two allowed transitions are $m_g=m_e=\pm 1/2$, givingFor propagation perpendicular to $z$ with linear polarization also perpendicular to $z$, the optical electric field is an equal-amplitude combination of the two field-axis circular components. It drives ${\sigma }^{\pm }$ transitions with $\Delta m=\pm 1$. These connect $-1/2\to +1/2$ and $+1/2\to -1/2$, givingThus each orientation selects two lines, with twice the separation in the perpendicular-polarization case. Neither pair has an unshifted central component. This is polarization-selective Zeeman splitting of an alkali D1 line, an anomalous Zeeman effect rather than a normal orbital triplet.