Chapter 21 Quiz

20 questions testing conceptual understanding and computational fluency. Answers in Appendix H.

Q1. In the interaction picture, the state vector $|\psi_I(t)\rangle$ evolves according to:

(a) $i\hbar\,d|\psi_I\rangle/dt = \hat{H}_0|\psi_I\rangle$ (b) $i\hbar\,d|\psi_I\rangle/dt = (\hat{H}_0 + \hat{V})|\psi_I\rangle$ (c) $i\hbar\,d|\psi_I\rangle/dt = \hat{V}_I(t)|\psi_I\rangle$ (d) $|\psi_I\rangle$ is time-independent

Q2. The interaction-picture perturbation $\hat{V}_I(t) = e^{i\hat{H}_0 t/\hbar}\hat{V}(t)e^{-i\hat{H}_0 t/\hbar}$ differs from $\hat{V}(t)$ because:

(a) It is multiplied by a scalar phase factor (b) Its matrix elements between energy eigenstates acquire oscillating phases $e^{i\omega_{fn}t}$ (c) It is no longer Hermitian (d) It changes the eigenvalues of $\hat{H}_0$

Q3. For a constant perturbation $\hat{V}$ turned on at $t = 0$, the first-order transition probability $P_{i \to f}(t)$ is proportional to:

(a) $t$ (b) $t^2$ at exact resonance ($E_f = E_i$) (c) $e^{-t}$ (d) $\cos^2(\omega_{fi} t)$

Q4. The first-order transition probability for a constant perturbation, $P_{i\to f}(t) = (|V_{fi}|^2/\hbar^2)\sin^2(\omega_{fi}t/2)/(\omega_{fi}/2)^2$, is largest when:

(a) $E_f \gg E_i$ (b) $E_f = E_i$ (c) $E_f = E_i + 2\pi\hbar/t$ (d) $E_f = E_i - |V_{fi}|$

Q5. Fermi's golden rule gives a transition rate (probability per unit time) rather than an oscillating probability because:

(a) Higher-order terms in perturbation theory cancel the oscillations (b) The perturbation is assumed to vary randomly in time (c) The sum over a continuum of final states converts the oscillating $\sin^2$ into a linearly growing probability (d) The interaction picture eliminates all oscillating terms

Q6. In Fermi's golden rule $\Gamma = (2\pi/\hbar)|V_{fi}|^2\rho(E_f)$, the density of states $\rho(E_f)$ is evaluated at:

(a) $E_f = 0$ (b) $E_f = E_i$ (energy of the initial state) (c) $E_f = E_i + \hbar\omega$ (for a harmonic perturbation at frequency $\omega$) (d) Both (b) and (c), depending on the type of perturbation

Q7. The electric dipole approximation $e^{i\vec{k}\cdot\vec{r}} \approx 1$ is valid because:

(a) The photon momentum is exactly zero (b) The wavelength of light is much larger than the atom ($\lambda \gg a_0$) (c) The electric field is always uniform in space (d) The magnetic field of the radiation does not interact with electrons

Q8. The transition dipole moment $\vec{d}_{fi} = \langle f|q\hat{\vec{r}}|i\rangle$ determines the strength of an E1 transition. If $\vec{d}_{fi} = 0$, the transition is:

(a) Completely impossible by any mechanism (b) Forbidden in the E1 approximation but may occur via M1, E2, or multi-photon processes (c) Allowed only at very high temperatures (d) Allowed but with a rate independent of the radiation intensity

Q9. Einstein showed that the stimulated emission coefficient $B_{21}$ and the absorption coefficient $B_{12}$ satisfy:

(a) $B_{21} = 2B_{12}$ (b) $B_{21} = B_{12}$ (they are equal) (c) $B_{21} = (g_1/g_2)B_{12}$ where $g$ are degeneracies (d) $B_{21}/B_{12} = e^{-\hbar\omega/k_BT}$

Q10. The spontaneous emission rate $A_{21}$ scales with the transition frequency as:

(a) $A \propto \omega$ (b) $A \propto \omega^2$ (c) $A \propto \omega^3$ (d) $A \propto \omega^4$

Q11. The hydrogen $2s$ state has a lifetime of $\sim 0.14$ s, while the $2p$ state has a lifetime of $\sim 1.6$ ns. This enormous difference is because:

(a) The $2s$ state has higher energy than the $2p$ state (b) The $2s \to 1s$ transition is E1-forbidden ($\Delta l = 0$) (c) The $2s$ state is not bound (d) Spontaneous emission is suppressed by destructive interference in the $2s$ state

Q12. For the electric dipole selection rule $\Delta l = \pm 1$, the physical reason is:

(a) The photon carries one unit of angular momentum, so the atom's orbital angular momentum must change by one unit (b) The atom must conserve energy during the transition (c) The electric field cannot couple states with the same parity (d) Both (a) and (c) — they are equivalent statements

Q13. Which transition is NOT E1-allowed in hydrogen?

(a) $3p \to 1s$ (b) $3d \to 2p$ (c) $3d \to 1s$ (d) $4f \to 3d$

Q14. Population inversion ($N_2 > N_1$) is necessary for laser action because:

(a) Stimulated emission requires $N_2 > N_1$ to outpace absorption (b) Spontaneous emission cannot occur without population inversion (c) The optical cavity requires more atoms in the upper state to resonate (d) Photons can only be emitted when $N_2 > N_1$

Q15. A four-level laser has a lower threshold than a three-level laser because:

(a) Four-level systems have stronger transition dipole moments (b) The lower laser level is rapidly depopulated, so $N_1 \approx 0$ and any $N_2 > 0$ gives inversion (c) Four-level systems have a higher density of states (d) The fourth level provides additional photon emission

Q16. Laser light differs from thermal light primarily in its:

(a) Frequency (lasers operate at higher frequencies) (b) Temporal and spatial coherence (laser photons are in the same mode) (c) Total energy (lasers always emit more total energy) (d) Polarization (laser light is always circularly polarized)

Q17. In Fermi's golden rule, what happens to the transition rate $\Gamma$ if the perturbation matrix element $|V_{fi}|$ is doubled?

(a) $\Gamma$ doubles (b) $\Gamma$ quadruples (c) $\Gamma$ increases by $\sqrt{2}$ (d) $\Gamma$ remains unchanged (only $\rho(E_f)$ matters)

Q18. The rotating wave approximation (RWA) in the context of a harmonic perturbation $\hat{V}(t) = \hat{V}_0\cos\omega t$ consists of:

(a) Averaging the perturbation over one period (b) Keeping only the near-resonant term and dropping the far-off-resonant (counter-rotating) term (c) Replacing $\cos\omega t$ with 1 (d) Setting $\omega = \omega_{fi}$ exactly

Q19. The 21-cm hydrogen line has an Einstein A coefficient of about $2.87 \times 10^{-15}\,\text{s}^{-1}$, corresponding to a lifetime of $\sim 11$ million years. This extremely long lifetime is primarily because:

(a) The hydrogen atom is very stable (b) The 21-cm transition is a magnetic dipole transition at radio frequencies, and $A \propto \omega^3$ (c) The transition is between different principal quantum numbers (d) Interstellar hydrogen is very cold, suppressing spontaneous emission

Q20. A photon at the resonance frequency $\omega_{fi}$ passes through a medium with more atoms in the lower state ($N_1 > N_2$). What happens to the photon?

(a) It is always absorbed (b) It is more likely to be absorbed than to stimulate emission (net absorption) (c) It passes through unaffected (d) It stimulates emission with certainty