Chapter 21 Exercises

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 21 Exercises

28 problems spanning computation, analysis, and conceptual reasoning. Difficulty ratings: ★ (routine), ★★ (intermediate), ★★★ (challenging), ★★★★ (advanced/research-level).

Section 21.1–21.2: Interaction Picture and First-Order Transitions

Problem 21.1 ★ A system starts in energy eigenstate $|1\rangle$ of $\hat{H}_0$, with eigenvalues $E_1$ and $E_2$ for states $|1\rangle$ and $|2\rangle$. (a) Write the general interaction-picture state $|\psi_I(t)\rangle$ in terms of the expansion coefficients $c_1(t)$ and $c_2(t)$. (b) Write the coupled differential equations for $c_1(t)$ and $c_2(t)$ given a perturbation $\hat{V}(t)$ with matrix elements $V_{12}(t) = V_{21}^*(t)$. (c) What is the initial condition at $t = 0$? (d) Show that $|c_1(t)|^2 + |c_2(t)|^2 = 1$ for all $t$ (probability conservation), assuming the exact equations are solved.

Problem 21.2 ★ Verify that the interaction-picture perturbation $\hat{V}_I(t) = e^{i\hat{H}_0 t/\hbar}\hat{V}(t)e^{-i\hat{H}_0 t/\hbar}$ satisfies: (a) $\hat{V}_I(t) = \hat{V}(t)$ when $[\hat{H}_0, \hat{V}(t)] = 0$ for all $t$. (b) The matrix elements $\langle f|\hat{V}_I(t)|i\rangle = e^{i\omega_{fi}t}V_{fi}(t)$. (c) If $\hat{V}$ is Hermitian, then $\hat{V}_I$ is Hermitian.

Problem 21.3 ★★ A two-level system with $E_1 = 0$ and $E_2 = \hbar\omega_0$ is subject to a constant perturbation turned on at $t = 0$: $V_{12} = V = \text{const}$. (a) Calculate the first-order transition probability $P_{1\to 2}(t)$. (b) At what times does $P_{1\to 2}$ reach its maximum values? (c) Show that for $|V| \ll \hbar\omega_0$, the first-order result agrees with the weak-field limit of the exact Rabi formula $P_{1\to 2}(t) = (\Omega_R^2/\Omega^2)\sin^2(\Omega t/2)$, where $\Omega_R = 2|V|/\hbar$ and $\Omega = \sqrt{\omega_0^2 + \Omega_R^2}$.

Problem 21.4 ★★ A particle in its ground state $|0\rangle$ is subject to a perturbation $\hat{V}(t) = \hat{V}_0 e^{-t/\tau}$ for $t > 0$, where $\hat{V}_0$ is time-independent and $\tau$ is a decay constant. (a) Calculate the first-order transition amplitude $c_f^{(1)}(t)$ to an excited state $|f\rangle$ with $\omega_{f0} = (E_f - E_0)/\hbar$. (b) Take the limit $t \to \infty$ and find $P_{0 \to f}(\infty)$. (c) Show that the transition probability is a Lorentzian centered at $\omega_{f0} = 0$ with width $1/\tau$. (d) Compare with the sudden ($\tau \to 0$) and adiabatic ($\tau \to \infty$) limits.

Problem 21.5 ★★ A harmonic oscillator in its ground state $|0\rangle$ is subjected to a time-dependent perturbation $\hat{V}(t) = \lambda \hat{x}\, e^{-|t|/\tau}$ (turned on slowly from $t = -\infty$, peaked at $t = 0$, and dying away). (a) Calculate the first-order transition amplitude to state $|n\rangle$ by integrating from $-\infty$ to $+\infty$. (b) Using $\langle n|\hat{x}|0\rangle = \sqrt{\hbar/(2m\omega)}\,\delta_{n,1}$, show that only the transition $|0\rangle \to |1\rangle$ is nonzero at first order. (c) Find $P_{0\to 1}$ as a function of $\omega\tau$. For what value of $\tau$ is the transition probability maximized?

Problem 21.6 ★★★ Consider the second-order correction to the transition amplitude:

$$c_f^{(2)}(t) = \left(-\frac{i}{\hbar}\right)^2 \sum_m \int_0^t dt' \int_0^{t'} dt''\, V_{fm}(t')e^{i\omega_{fm}t'} V_{mi}(t'')e^{i\omega_{mi}t''}$$

(a) Explain physically what this expression represents (transitions via intermediate states $|m\rangle$). (b) For a constant perturbation, show that the second-order contribution to the transition rate involves a sum over intermediate virtual states. (c) Under what conditions does second-order perturbation theory become important? Give a physical example.

Section 21.3: Fermi's Golden Rule

Problem 21.7 ★★ Prove that the function $f_t(\omega) = \sin^2(\omega t/2)/(\omega/2)^2$ satisfies: (a) $f_t(0) = t^2$ (the peak height). (b) The first zero is at $\omega = 2\pi/t$ (the peak width scales as $1/t$). (c) $\int_{-\infty}^{\infty} f_t(\omega)\,d\omega = 2\pi t$. (d) Therefore $\lim_{t\to\infty} f_t(\omega)/(2\pi t) = \delta(\omega)$.

Problem 21.8 ★★ A particle in a one-dimensional box of length $L$ is in its ground state. A constant perturbation $\hat{V} = V_0\sin(\pi x/L)$ is turned on at $t = 0$. Using Fermi's golden rule: (a) Calculate the matrix elements $\langle n|\hat{V}|1\rangle$. (b) Why does Fermi's golden rule not directly apply here (discrete spectrum)? What would need to change for it to apply? (c) If the box is very large ($L \to \infty$) so that the spectrum becomes quasi-continuous, estimate the transition rate using the density of states for a particle in a 1D box.

Problem 21.9 ★★★ A free particle with momentum $\vec{p}_i$ scatters off a localized potential $V(\vec{r})$. Using Fermi's golden rule: (a) Show that the transition rate to a final state with momentum $\vec{p}_f$ involves the Fourier transform of $V(\vec{r})$: $\tilde{V}(\vec{q}) = \int V(\vec{r})\,e^{-i\vec{q}\cdot\vec{r}}\,d^3r$, where $\vec{q} = (\vec{p}_f - \vec{p}_i)/\hbar$. (b) Show that the density of final states in 3D is $\rho(E_f) = (m L^3)/(2\pi^2\hbar^3)\sqrt{2mE_f}$ (using box normalization with volume $L^3$). (c) Derive the differential scattering cross section $d\sigma/d\Omega = (m/(2\pi\hbar^2))^2|\tilde{V}(\vec{q})|^2$. (d) Verify that this is the Born approximation (Chapter 22 preview).

Problem 21.10 ★★ An atom initially in state $|i\rangle$ is irradiated by a short pulse of radiation lasting from $t = 0$ to $t = T$, with electric field $\vec{E}(t) = \vec{E}_0\cos(\omega t)$ for $0 \leq t \leq T$ and zero otherwise. (a) Calculate the first-order transition probability $P_{i \to f}(T)$ for a final state $|f\rangle$ with $E_f > E_i$. (b) Show that for a long pulse ($\omega_{fi}T \gg 1$), the result reduces to the Fermi golden rule rate times $T$. (c) For a very short pulse ($T \to 0$ with $E_0 T$ held fixed), show that many final states are excited (broad energy distribution).

Problem 21.11 ★★★ Fermi's golden rule for a harmonic perturbation $\hat{V}(t) = \hat{V}_0\cos\omega t$ gives a transition rate $\Gamma(\omega)$ as a function of the driving frequency $\omega$. (a) Starting from the first-order probability at finite time $t$, show that the "lineshape" (before taking $t \to \infty$) is $\propto \sin^2[(\omega_{fi} - \omega)t/2]/(\omega_{fi} - \omega)^2$. (b) The "natural linewidth" of a transition is $\Delta\omega \sim \Gamma$ (due to the finite lifetime $\tau = 1/\Gamma$ of the excited state). Show that this corresponds to a Lorentzian lineshape $I(\omega) \propto \Gamma/[(\omega - \omega_{fi})^2 + (\Gamma/2)^2]$ in the Wigner-Weisskopf theory (state the result — full derivation not required). (c) Estimate the natural linewidth of the hydrogen Lyman-$\alpha$ line ($A = 6.27 \times 10^8\,\text{s}^{-1}$). Express in Hz and in eV.

Section 21.4–21.5: Electromagnetic Transitions

Problem 21.12 ★ Starting from the minimal coupling Hamiltonian, show that: (a) In the Coulomb gauge ($\nabla\cdot\vec{A} = 0$), $\hat{\vec{p}}\cdot\vec{A} = \vec{A}\cdot\hat{\vec{p}}$. (b) The perturbation for an electron in an electromagnetic field is $\hat{V} = (e/m_e c)\,\vec{A}\cdot\hat{\vec{p}}$ (CGS) or $\hat{V} = (e/m_e)\,\vec{A}\cdot\hat{\vec{p}}$ (SI). (c) The $\vec{A}\cdot\vec{A}$ term can be neglected when the number of photons per mode is much less than $(m_e c/e)^2/\hbar\omega$.

Problem 21.13 ★★ Derive the identity $\langle f|\hat{\vec{p}}|i\rangle = im\omega_{fi}\langle f|\hat{\vec{r}}|i\rangle$ (the "length-velocity equivalence"): (a) Start from the commutator $[\hat{x}_j, \hat{H}_0] = i\hbar\hat{p}_j/m$ (valid when $\hat{H}_0 = \hat{p}^2/2m + V(\hat{\vec{r}})$). (b) Take the matrix element $\langle f|[\hat{x}_j, \hat{H}_0]|i\rangle$ and use $\hat{H}_0|i\rangle = E_i|i\rangle$. (c) Explain why this identity fails if $\hat{H}_0$ contains velocity-dependent terms (e.g., spin-orbit coupling).

Problem 21.14 ★★ Evaluate the electric dipole matrix elements for hydrogen: (a) Show that the matrix element $\langle n'l'm'|r\cos\theta|nlm\rangle$ is nonzero only if $\Delta l = \pm 1$ and $\Delta m = 0$ by using the relation $\cos\theta = \sqrt{4\pi/3}\,Y_1^0$. (b) Calculate $|\langle 2,1,0|r\cos\theta|1,0,0\rangle|$ for hydrogen. Express the result in units of $ea_0$. (c) Repeat for $\langle 2,1,\pm 1|r\sin\theta\,e^{\pm i\phi}|1,0,0\rangle$ and show it gives the same $|\vec{d}_{fi}|^2$ when summed over polarizations.

Problem 21.15 ★★★ Calculate the spontaneous emission rate for the hydrogen $2p \to 1s$ transition from first principles: (a) Write out the radial wave functions $R_{10}(r)$ and $R_{21}(r)$ for hydrogen. (b) Compute the radial integral $\int_0^\infty R_{21}(r)\, r\, R_{10}(r)\, r^2\, dr$. (c) Combine with the angular integral (Problem 21.14) to get $|\vec{d}_{21}|^2$, summing over all three $m$ sublevels of the $2p$ state. (d) Use the Einstein $A$ coefficient formula to compute the spontaneous emission rate. Verify that $A_{2p\to 1s} \approx 6.27 \times 10^8\,\text{s}^{-1}$ and the lifetime is $\tau \approx 1.60$ ns.

Problem 21.16 ★★ An electron in the $n = 2$, $l = 1$, $m = 0$ state of hydrogen is placed in a uniform electric field $\vec{E} = E_0\hat{z}$ that is turned on at $t = 0$. (a) To which states can it make E1 transitions? (List all allowed $n'l'm'$.) (b) Calculate the transition probability $P_{210 \to 100}(t)$ using first-order perturbation theory, treating the static field as the perturbation $\hat{V} = eE_0\hat{z}$. (c) Is this a "real" transition (with energy change) or a "virtual" transition? Explain the distinction.

Section 21.6: Spontaneous Emission and Einstein Coefficients

Problem 21.17 ★★ Reproduce Einstein's 1917 argument: (a) Write the rate equations for populations $N_1$ and $N_2$ of a two-level atom in thermal radiation, including absorption ($B_{12}u$), stimulated emission ($B_{21}u$), and spontaneous emission ($A_{21}$). (b) Impose thermal equilibrium ($dN_1/dt = dN_2/dt = 0$) and the Boltzmann distribution $N_2/N_1 = e^{-\hbar\omega_0/k_BT}$. (c) Solve for $u(\omega_0)$ and compare with the Planck spectrum to derive $B_{12} = B_{21}$ and $A_{21} = (\hbar\omega_0^3/\pi^2c^3)B_{21}$. (d) Explain why Einstein's argument works without quantizing the electromagnetic field.

Problem 21.18 ★★★ The spontaneous emission rate scales as $\omega^3$. Explore the consequences: (a) Calculate the ratio of spontaneous emission rates for the hydrogen $3p \to 1s$ ($\lambda = 102.6$ nm) and $3p \to 2s$ ($\lambda = 656.3$ nm) transitions, assuming the dipole matrix elements have comparable magnitudes. (b) Show that for the 21-cm hyperfine line of hydrogen ($\nu = 1420$ MHz), the Einstein $A$ coefficient is $A \approx 2.87 \times 10^{-15}\,\text{s}^{-1}$, corresponding to a lifetime of $\sim 11$ million years. (c) Despite this incredibly slow rate, the 21-cm line is easily observable in radio astronomy. Explain why (consider the amount of hydrogen in a typical galaxy).

Problem 21.19 ★★ For degenerate energy levels (as in hydrogen, where the $2p$ level has three $m$ substates): (a) Show that the total spontaneous emission rate from all $m$ substates of the $2p$ level to the $1s$ level is $A_{\text{total}} = \sum_{m=-1}^{+1} A_{2p,m \to 1s} = 3A_{2p,0 \to 1s}$ (when summing over all polarizations). (b) Prove that this result follows from the isotropy of space: the total rate cannot depend on the quantization axis. (c) More generally, for a transition from an upper level with angular momentum $J$ to a lower level with angular momentum $J'$, the average rate per sublevel is $A/(2J + 1)$ where $A$ is the total rate. Explain why.

Problem 21.20 ★★★★ (Research-level) Consider the ratio of spontaneous to stimulated emission rates in several astrophysical environments: (a) The surface of the Sun ($T \approx 5778$ K) at $\lambda = 500$ nm. (b) The cosmic microwave background ($T = 2.725$ K) at $\lambda = 2$ mm (the peak of the CMB spectrum). (c) The interior of a laser cavity where the photon number per mode is $\bar{n} \approx 10^{12}$. (d) Explain qualitatively why stimulated emission is negligible in most natural light sources but is the dominant process inside lasers.

Section 21.7: Selection Rules

Problem 21.21 ★★ Determine which of the following hydrogen transitions are E1-allowed. For those that are forbidden, state whether they are allowed as M1 or E2 transitions. (a) $3d \to 2p$ (b) $3d \to 1s$ (c) $3s \to 1s$ (d) $4f \to 2p$ (e) $4f \to 3d$ (f) $4d \to 2s$

Problem 21.22 ★★ Selection rules for the harmonic oscillator: (a) Show that the matrix element $\langle m|\hat{x}|n\rangle = \sqrt{\hbar/(2m\omega)}(\sqrt{n+1}\,\delta_{m,n+1} + \sqrt{n}\,\delta_{m,n-1})$ using ladder operators. (b) Conclude that the electric dipole selection rule for the harmonic oscillator is $\Delta n = \pm 1$. (c) Show that the $\hat{x}^2$ matrix element (relevant for E2 or Raman transitions) allows $\Delta n = 0, \pm 2$. (d) Real molecular vibrations are anharmonic. How does anharmonicity modify the selection rules? What are "overtone" transitions?

Problem 21.23 ★★★ Selection rules for multi-electron atoms in LS coupling: (a) Explain why $\Delta S = 0$ for electric dipole transitions (the dipole operator $e\hat{\vec{r}}$ does not act on spin). (b) The transition ${}^1S_0 \to {}^3S_1$ is doubly forbidden ($\Delta S = 1$ and $\Delta L = 0$). In what physical situations might such a transition still occur? (c) The ${}^1S_0 \to {}^3P_1$ transition is forbidden by $\Delta S = 0$ in pure LS coupling, but is observed in heavy atoms like mercury ($\lambda = 253.7$ nm). Explain why (consider spin-orbit mixing). (d) How do the selection rules change in $jj$-coupling (relevant for very heavy atoms)?

Section 21.8: Lasers

Problem 21.24 ★★ Laser rate equations: Consider a four-level laser with levels 0 (ground), 1 (lower laser), 2 (upper laser), 3 (pump). (a) Write the rate equation for the population $N_2$ of the upper laser level, including pump rate $R_p$, spontaneous emission $A_{21}$, and stimulated emission $B_{21}\rho(\omega)$. (b) In steady state ($dN_2/dt = 0$), solve for $N_2$ as a function of $R_p$ and $A_{21}$. (c) Find the threshold pump rate $R_p^{\text{th}}$ for population inversion ($N_2 > N_1$), assuming level 1 is rapidly depopulated ($N_1 \approx 0$). Compare with a three-level system where level 1 is the ground state.

Problem 21.25 ★★★ The gain coefficient of a laser medium is $g(\omega) = (A_{21}\lambda^2/8\pi)(N_2 - N_1)\,\mathcal{L}(\omega - \omega_0)$, where $\mathcal{L}$ is a normalized Lorentzian lineshape function. (a) Show that $g > 0$ requires population inversion ($N_2 > N_1$). (b) For a cavity of length $L$ with mirror reflectivities $R_1$ and $R_2$, the threshold condition is $R_1 R_2 e^{2gL} = 1$. Derive the minimum population inversion needed for lasing. (c) A HeNe laser has $L = 30$ cm, $R_1 = 1.0$, $R_2 = 0.98$, $A_{21} = 3.4 \times 10^6\,\text{s}^{-1}$, $\lambda = 632.8$ nm, and linewidth $\Delta\nu = 1.5$ GHz. Estimate the threshold population inversion $\Delta N = N_2 - N_1$.

Problem 21.26 ★★ Laser linewidth: An ideal single-mode laser has a linewidth given by the Schawlow-Townes formula:

$$\Delta\nu_{\text{laser}} = \frac{2\pi h\nu(\Delta\nu_c)^2}{P_{\text{out}}}$$

where $\Delta\nu_c$ is the cavity linewidth and $P_{\text{out}}$ is the output power. (a) For a HeNe laser with $\Delta\nu_c = 5$ MHz, $\nu = 4.74 \times 10^{14}$ Hz, $P = 1$ mW, calculate $\Delta\nu_{\text{laser}}$. (b) Express this as a fractional frequency stability $\Delta\nu/\nu$. (c) Compare with the natural linewidth of the $3s_2 \to 2p_4$ neon transition ($A \approx 3.4 \times 10^6\,\text{s}^{-1}$). Why is the laser linewidth so much narrower?

Synthesis and Challenge Problems

Problem 21.27 ★★★ The "sum rule" (Thomas-Reiche-Kuhn / f-sum rule): Define the oscillator strength for the transition $|i\rangle \to |f\rangle$:

$$f_{fi} = \frac{2m\omega_{fi}}{3\hbar}\,|\langle f|\hat{\vec{r}}|i\rangle|^2$$

(a) Show that $f_{fi}$ is dimensionless. (b) Prove the sum rule $\sum_f f_{fi} = 1$ (the sum is over all final states) by evaluating $\sum_f f_{fi}$ using the completeness relation and the commutator $[\hat{x}, [\hat{x}, \hat{H}_0]] = \hbar^2/m$. (c) Explain the physical meaning of this result: the total "absorption strength" of any quantum system equals the classical value (one electron oscillator). (d) For hydrogen in the ground state, the $1s \to 2p$ transition has $f = 0.416$. What fraction of the total oscillator strength is "used up" by this single transition?

Problem 21.28 ★★★★ (Research-level) Multi-photon transitions and beyond first order: When a single-photon transition is forbidden by selection rules, two-photon absorption can occur (the system absorbs two photons simultaneously, each of energy $\hbar\omega \approx \Delta E/2$).

(a) Using second-order perturbation theory, show that the two-photon transition rate is proportional to $I^2$ (intensity squared), in contrast to single-photon absorption which is proportional to $I$.

(b) The selection rules for two-photon transitions are $\Delta l = 0, \pm 2$ (the opposite of E1!). Explain qualitatively why (each photon changes $l$ by $\pm 1$, and two photons are absorbed).

(c) Two-photon absorption is used in fluorescence microscopy to achieve sub-diffraction-limit resolution. The key is the $I^2$ dependence. Explain why this helps with resolution (hint: $I^2$ falls off faster away from the focal point than $I$).

(d) Estimate the ratio of two-photon to one-photon absorption rates for typical laser intensities ($I \sim 10^{10}\,\text{W/cm}^2$) at optical frequencies.