GATE Physics Atomic and Molecular Physics Syllabus

Write the Landé g-factor formula for LS coupling.

$$g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$$

What is the energy shift of a level with magnetic quantum number $m_J$ in the (weak-field) anomalous Zeeman effect?

$$\Delta E = g_J\, \mu_B\, B\, m_J$$ where $\mu_B = \frac{e\hbar}{2m_e}$ is the Bohr magneton.

What is the Paschen-Back effect?

The strong-field limit of the Zeeman effect, where the external magnetic field is strong enough to decouple $\vec{L}$ and $\vec{S}$ (spin-orbit coupling broken). $\vec{L}$ and $\vec{S}$ then precess independently about $\vec{B}$, with energy $\Delta E = (m_L + 2m_S)\mu_B B$.

What is the Stark effect?

The shifting and splitting of atomic spectral lines due to an external static electric field, the electric-field analogue of the Zeeman effect.

Distinguish the linear and quadratic Stark effects.

Linear (first-order) Stark effect: energy shift $\propto E$, occurs in hydrogen and degenerate states having a permanent dipole moment (states of mixed parity). Quadratic (second-order) Stark effect: shift $\propto E^2$, occurs in non-degenerate states (most atoms) via an induced dipole.

Why does hydrogen show a linear Stark effect while most atoms show only quadratic?

Hydrogen has degenerate states of different $l$ (same $n$) and opposite parity that can be mixed by the field to produce a permanent dipole, giving a first-order shift. Non-degenerate states have no permanent dipole, so the leading effect is the quadratic (induced-dipole) term.

What are electric dipole (E1) transitions?

Radiative transitions driven by the electric dipole operator $\vec{d} = -e\vec{r}$. They are the strongest (most probable) radiative transitions and dominate atomic spectra; their probability depends on the dipole matrix element $\langle f|\vec{r}|i\rangle$.

State the electric dipole selection rules for $l$ and $m_l$ (single electron).

$\Delta l = \pm 1$ and $\Delta m_l = 0, \pm 1$. The parity must change.

State the electric dipole selection rules for $L$, $S$, and $J$ in LS coupling.

$\Delta L = 0, \pm 1$ (but $L=0 \nrightarrow L=0$); $\Delta S = 0$; $\Delta J = 0, \pm 1$ (but $J=0 \nrightarrow J=0$); parity changes.

What is the parity (Laporte) selection rule for E1 transitions?

The Laporte rule: electric dipole transitions occur only between states of opposite parity (parity must change), because the dipole operator $\vec{r}$ is odd under inversion.

State the hyperfine selection rule on $F$ for electric dipole transitions.

$\Delta F = 0, \pm 1$, with $F=0 \nrightarrow F=0$ forbidden.

What is a forbidden transition, and how can it still occur?

A transition violating the E1 selection rules (e.g., parity-conserving). It can still occur via higher-order processes such as magnetic dipole (M1) or electric quadrupole (E2) radiation, which are much weaker, giving long-lived metastable states.

State the Franck-Condon principle.

Electronic transitions in molecules occur so rapidly (compared to nuclear motion) that the internuclear distance and nuclear momenta are essentially unchanged during the transition. Hence transitions are 'vertical' on a potential-energy-vs-internuclear-distance diagram.

What is the Franck-Condon factor?

The square of the overlap integral between the vibrational wavefunctions of the initial and final electronic states, $|\langle \chi_{v'} | \chi_{v''}\rangle|^2$. It governs the relative intensities of vibronic (vibrational-electronic) bands.

On a potential-energy diagram, which vibronic transition is most intense according to the Franck-Condon principle?

The transition is most intense to the vibrational level of the upper state whose wavefunction has maximum overlap with the lower state's wavefunction — typically a vertical transition reaching the turning point of the upper-state potential directly above the lower-state equilibrium position.

Define Einstein's coefficient $A_{21}$.

$A_{21}$ is the rate (probability per unit time) of spontaneous emission from upper level 2 to lower level 1. The spontaneous decay rate is $\frac{dN_2}{dt} = -A_{21}N_2$, and $1/A_{21}$ is the radiative lifetime.

Define Einstein's coefficients $B_{12}$ and $B_{21}$.

$B_{12}$ is the absorption coefficient: rate of $1\to2$ transitions per unit spectral energy density, $\propto B_{12}\rho(\nu)$. $B_{21}$ is the stimulated-emission coefficient: rate of induced $2\to1$ emission, $\propto B_{21}\rho(\nu)$.

State the relation between the Einstein B coefficients including degeneracies.

$$g_1 B_{12} = g_2 B_{21}$$ where $g_1, g_2$ are the degeneracies of the lower and upper levels. For non-degenerate levels $B_{12} = B_{21}$.

State the relation between Einstein's $A_{21}$ and $B_{21}$ coefficients.

$$\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$$ This shows spontaneous emission grows as $\nu^3$, dominating at high frequencies (hard to build X-ray lasers).

How is the ratio of spontaneous to stimulated emission rates related to temperature?

$$\frac{A_{21}}{B_{21}\rho(\nu)} = e^{h\nu/k_BT} - 1.$$ Stimulated emission dominates when $h\nu \ll k_BT$; spontaneous emission dominates when $h\nu \gg k_BT$.

What is population inversion?

A non-equilibrium condition in which the population of a higher energy level exceeds that of a lower level, $N_2 > N_1$ (or $N_2/g_2 > N_1/g_1$). It is the necessary condition for net stimulated emission (optical gain / lasing).

Why is population inversion impossible at thermal equilibrium?

In thermal equilibrium populations follow the Boltzmann distribution $\frac{N_2}{N_1} = e^{-(E_2-E_1)/k_BT} < 1$, so the upper level is always less populated; achieving $N_2 > N_1$ requires external 'pumping' (a non-equilibrium process).

Why can a two-level system not achieve population inversion by optical pumping alone?

In a two-level system, absorption ($B_{12}$) and stimulated emission ($B_{21}$) balance; the most pumping can do is equalize populations ($N_1 = N_2$, saturation/transparency). Net inversion $N_2 > N_1$ is impossible with optical pumping alone.

Describe the operation of a three-level laser system.

Atoms are pumped from ground level 1 to a higher level 3, then decay rapidly (non-radiatively) to a metastable level 2. Lasing occurs on the $2\to1$ transition. Inversion is between level 2 and the ground state 1.

What is the main disadvantage of a three-level laser compared with a four-level laser?

In a three-level laser the lower laser level is the ground state, which is heavily populated, so more than half the atoms must be pumped to level 2 to achieve inversion — requiring a high pump threshold. The ruby laser is the classic example.

Why is a four-level laser more efficient than a three-level laser?

Its lower laser level is an excited state that is essentially empty (rapidly depopulated to the ground state), so even a small upper-level population gives inversion. This means a much lower pumping threshold and continuous operation are possible.

What is a metastable state and why is it important for lasers?

A metastable state is an excited level with an unusually long lifetime (transitions to lower states are forbidden by E1 selection rules). It allows atoms to accumulate, building up the population inversion needed for lasing.

What is the threshold condition for laser oscillation (round-trip gain)?

The round-trip optical gain must at least equal the round-trip losses: the gain coefficient satisfies $R_1R_2 e^{2(g-\alpha)L} \geq 1$, i.e., gain from stimulated emission compensates mirror and cavity losses. Equivalently the population inversion exceeds a threshold value $\Delta N_{th}$.

Compare the selection rules and relative strengths of E1, M1, and E2 transitions.

E1 (electric dipole): parity changes, $\Delta l = \pm1$, strongest. M1 (magnetic dipole): parity unchanged, $\Delta l = 0$, $\sim \alpha^2$ weaker than E1. E2 (electric quadrupole): parity unchanged, $\Delta l = 0,\pm2$, also $\sim \alpha^2$ weaker than E1.

What total angular momentum $F$ values arise for a level with $J = 1/2$ and nuclear spin $I = 3/2$?

$F$ ranges from $|I-J|$ to $I+J$: $F = 1$ and $F = 2$, giving two hyperfine levels.