Chapter 7 Exercises

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 7 Exercises

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

Section 7.1–7.2: Time-Evolution Operator and Stationary States

Problem 7.1 ★ A particle is in the energy eigenstate $\psi_3(x)$ of an infinite square well of width $a$. (a) Write the full time-dependent state $\Psi_3(x,t)$. (b) Calculate $|\Psi_3(x,t)|^2$ and verify it is time-independent. (c) Calculate $\langle \hat{x} \rangle(t)$ and $\langle \hat{p} \rangle(t)$. Are they time-independent? Should they be?

Problem 7.2 ★ Verify explicitly that $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ satisfies: (a) The initial condition $\hat{U}(0) = \hat{I}$ (b) The composition law $\hat{U}(t_1 + t_2) = \hat{U}(t_1)\hat{U}(t_2)$ when $\hat{H}$ is time-independent (c) Unitarity: $\hat{U}^\dagger(t)\hat{U}(t) = \hat{I}$ when $\hat{H} = \hat{H}^\dagger$

Problem 7.3 ★★ A photon enters a Mach-Zehnder interferometer. The upper arm has a glass plate of thickness $d$ and refractive index $n$, while the lower arm is unobstructed. (a) Calculate the phase difference $\Delta\phi$ between the two arms. (b) For $n = 1.5$, $d = 632.8$ nm (one wavelength of HeNe laser), find the detection probability at each output port. (c) At what thickness $d$ is all the light directed to a single output?

Problem 7.4 ★★ Show that for a time-independent Hamiltonian, the time-evolution operator has the spectral decomposition:

$$\hat{U}(t) = \sum_n e^{-iE_n t/\hbar}\, |\psi_n\rangle\langle\psi_n|$$

where $\hat{H}|\psi_n\rangle = E_n|\psi_n\rangle$. Use this to find $\hat{U}(t)$ explicitly for a two-level system with $\hat{H} = E_1|1\rangle\langle 1| + E_2|2\rangle\langle 2|$.

Problem 7.5 ★★★ (a) Show that if $\hat{H}(t_1)$ and $\hat{H}(t_2)$ commute for all $t_1, t_2$, then the time-ordered exponential reduces to the ordinary exponential $\hat{U}(t) = \exp\left(-\frac{i}{\hbar}\int_0^t \hat{H}(t')dt'\right)$. (b) Give a physical example where $\hat{H}(t_1)$ and $\hat{H}(t_2)$ do commute. (c) Give a physical example where they do not.

Section 7.3: Wave Packets

Problem 7.6 ★ A Gaussian wave packet has initial width $\sigma_0 = 1$ nm and zero average momentum ($k_0 = 0$). Calculate the width at: (a) $t = 1$ fs (b) $t = 1$ ps (c) $t = 1$ ns for (i) an electron and (ii) a proton. At what time has the packet doubled in width for each particle?

Problem 7.7 ★★ A free-particle wave packet has the initial momentum distribution $\phi(k) = A e^{-\alpha|k-k_0|}$ (a Lorentzian-like distribution in $k$-space). (a) Normalize $\phi(k)$. (b) Find $\Psi(x, 0)$ by Fourier transform. (c) Find $\Psi(x, t)$ and $|\Psi(x, t)|^2$. Does this packet spread? Compare with the Gaussian case.

Problem 7.8 ★★ Show that the group velocity of a wave packet is $v_g = d\omega/dk|_{k_0}$ by expanding $\omega(k)$ about $k_0$ and identifying the center of the wave packet at time $t$. Under what conditions is the group velocity equal to the classical particle velocity $p/m$?

Problem 7.9 ★★★ A Gaussian wave packet moves in the potential $V(x) = mgx$ (uniform gravity). (a) Using Ehrenfest's theorem, find $\langle \hat{x}\rangle(t)$ and $\langle \hat{p}\rangle(t)$. (b) Show that the width $\sigma(t)$ of the packet evolves exactly as in the free-particle case — the gravitational field does not affect the spreading. (c) Explain physically why gravity does not cause additional spreading.

Section 7.4: Quantum Revivals

Problem 7.10 ★★ An electron is confined to an infinite square well of width $a = 10$ nm. (a) Calculate the revival time $T_{\text{rev}}$. (b) Calculate the classical period $T_{\text{cl}} = 2\pi\hbar/(E_2 - E_1)$ for a superposition of $n = 1$ and $n = 2$. (c) What is the ratio $T_{\text{rev}}/T_{\text{cl}}$?

Problem 7.11 ★★ Consider a particle in the state $\Psi(x, 0) = \frac{1}{\sqrt{2}}[\psi_1(x) + \psi_3(x)]$ in an infinite square well. (a) Write $\Psi(x, t)$ and find $|\Psi(x, t)|^2$. (b) At what times does the probability density return to its initial form? Is this the full revival time? (c) Compute $\langle \hat{x}\rangle(t)$. Does it oscillate? At what frequency?

Problem 7.12 ★★★ For the QHO with $E_n = (n + 1/2)\hbar\omega$: (a) Show that the revival time is $T_{\text{QHO}} = 2\pi/\omega$ — the classical oscillation period. (b) Explain why the QHO revival time equals the classical period, while for the infinite well, $T_{\text{rev}} \gg T_{\text{cl}}$. (c) Show that fractional revivals in the QHO are trivial (the wave packet does not split into copies). Why?

Problem 7.13 ★★★★ Revivals in the hydrogen atom. The hydrogen energy levels are $E_n = -E_0/n^2$ where $E_0 = 13.6$ eV. (a) Expand $E_n$ about a large value $\bar{n}$ to second order: $E_n \approx E_{\bar{n}} + \alpha(n - \bar{n}) + \beta(n - \bar{n})^2$. (b) Show that $T_{\text{cl}} = 2\pi\hbar/|\alpha|$ is the classical Kepler period for principal quantum number $\bar{n}$. (c) Show that $T_{\text{rev}} = 2\pi\hbar/|\beta|$ and compute $T_{\text{rev}}/T_{\text{cl}}$. (d) For $\bar{n} = 50$, calculate both $T_{\text{cl}}$ and $T_{\text{rev}}$ in seconds.

Section 7.5–7.7: Pictures of Quantum Mechanics

Problem 7.14 ★ Explicitly verify that the Schrödinger and Heisenberg pictures give the same expectation value $\langle \hat{x}\rangle(t)$ for the QHO ground state. (This should be straightforward — what should $\langle \hat{x}\rangle$ be for the ground state?)

Problem 7.15 ★★ In the Heisenberg picture, find $\hat{x}_H(t)$ and $\hat{p}_H(t)$ for a particle in a linear potential $V(x) = -Fx$ (constant force $F$). Verify that $\langle \hat{x}_H(t)\rangle$ matches the classical trajectory $x(t) = x_0 + (p_0/m)t + Ft^2/(2m)$.

Problem 7.16 ★★ Prove the Heisenberg equation of motion starting from the definition $\hat{A}_H(t) = \hat{U}^\dagger(t)\hat{A}_S\hat{U}(t)$. Show each step clearly, including where you use $i\hbar\partial_t\hat{U} = \hat{H}\hat{U}$.

Problem 7.17 ★★★ In the Heisenberg picture for the QHO: (a) Show that the ladder operators evolve as $\hat{a}_H(t) = \hat{a}(0)e^{-i\omega t}$ and $\hat{a}^\dagger_H(t) = \hat{a}^\dagger(0)e^{i\omega t}$. (b) Use (a) to derive $\hat{x}_H(t)$ and $\hat{p}_H(t)$ without solving differential equations. (c) Verify that $[\hat{x}_H(t), \hat{p}_H(t)] = i\hbar$ at all times $t$. Why must this be true?

Problem 7.18 ★★★ Consider a Hamiltonian $\hat{H} = \hat{H}_0 + \hat{V}$ where $\hat{H}_0$ has eigenstates $|n\rangle$ with energies $E_n$. (a) Show that the interaction-picture perturbation is $\hat{V}_I(t) = \sum_{m,n} V_{mn} e^{i(E_m - E_n)t/\hbar}|m\rangle\langle n|$. (b) For a two-level system, write out $\hat{V}_I(t)$ explicitly and identify the "slowly varying" and "rapidly varying" terms. This is the origin of the rotating wave approximation.

Problem 7.19 ★★ Classify each of the following problems as best suited for the Schrödinger, Heisenberg, or interaction picture. Justify your choice in one sentence for each. (a) Finding the energy eigenstates of the hydrogen atom (b) Computing the time evolution of a coherent state in the QHO (c) Calculating the transition rate for an atom absorbing a photon (d) Proving that angular momentum is conserved for a central potential (e) Deriving the S-matrix for electron-electron scattering

Section 7.8: Ehrenfest's Theorem

Problem 7.20 ★★ Derive the second Ehrenfest equation $d\langle\hat{p}\rangle/dt = -\langle dV/dx\rangle$ in full detail. Start from $\frac{d}{dt}\langle\hat{p}\rangle = \frac{i}{\hbar}\langle[\hat{H}, \hat{p}]\rangle$ and evaluate the commutator $[\hat{H}, \hat{p}]$ for $\hat{H} = \hat{p}^2/(2m) + V(\hat{x})$.

Problem 7.21 ★★ For the potential $V(x) = V_0 \cosh^{-2}(x/a)$ (Pöschl-Teller potential): (a) Compute $F(x) = -dV/dx$. (b) Show that $\langle F(\hat{x})\rangle \neq F(\langle\hat{x}\rangle)$ for a Gaussian wave packet of width $\sigma$ comparable to $a$. (c) Estimate the fractional correction $[\langle F\rangle - F(\langle x\rangle)]/F(\langle x\rangle)$ in terms of $\sigma/a$.

Problem 7.22 ★★★ Energy-time uncertainty from Ehrenfest. Starting from the generalized uncertainty principle (Chapter 6) $\Delta A \cdot \Delta H \geq \frac{1}{2}|\langle[\hat{A}, \hat{H}]\rangle|$: (a) Show that $\Delta A \cdot \Delta E \geq \frac{\hbar}{2}\left|\frac{d\langle\hat{A}\rangle}{dt}\right|$ for any observable $\hat{A}$. (b) Define the "lifetime" $\tau_A = \Delta A / |d\langle\hat{A}\rangle/dt|$ and show that $\Delta E \cdot \tau_A \geq \hbar/2$. (c) Apply this to the position operator for a free-particle wave packet and verify that the result is consistent with the spreading formula.

Section 7.9: Rabi Oscillations

Problem 7.23 ★ A two-level atom starts in state $|1\rangle$ and is driven at exact resonance ($\delta = 0$) with Rabi frequency $\Omega_R$. (a) At what time is the system in an equal superposition of $|1\rangle$ and $|2\rangle$? (b) At what time is the system completely in state $|2\rangle$? (c) What is the state at $t = 3\pi/\Omega_R$?

Problem 7.24 ★★ Derive the Rabi oscillation formula for the off-resonant case. Starting from the coupled equations after the RWA:

$$i\dot{c}_1 = \frac{\Omega_R}{2}e^{-i\delta t}c_2, \qquad i\dot{c}_2 = \frac{\Omega_R}{2}e^{i\delta t}c_1$$

with $c_1(0) = 1$, $c_2(0) = 0$: (a) Differentiate the first equation and substitute the second to get a second-order ODE for $c_1$. (b) Solve this ODE and find $c_1(t)$ and $c_2(t)$. (c) Show that $P_{1\to 2}(t) = (\Omega_R^2/\Omega^2)\sin^2(\Omega t/2)$ where $\Omega = \sqrt{\Omega_R^2 + \delta^2}$.

Problem 7.25 ★★ An NMR experiment uses a proton in a static magnetic field $B_0 = 2$ T with an RF pulse of amplitude $B_1 = 10^{-4}$ T at the Larmor frequency. (a) Calculate the Larmor frequency $\omega_0 = \gamma B_0$ using the proton gyromagnetic ratio $\gamma_p/(2\pi) = 42.58$ MHz/T. (b) Calculate the Rabi frequency $\Omega_R = \gamma B_1$. (c) Calculate the duration of a $\pi$-pulse and a $\pi/2$-pulse. (d) How many Larmor cycles occur during one Rabi cycle? This ratio justifies the RWA.

Problem 7.26 ★★★ Beyond the RWA. Consider the driven two-level system without making the rotating wave approximation. (a) Write the full coupled equations including the counter-rotating terms. (b) Using numerical methods (or refer to code/example-02-rabi.py), compare the exact solution with the RWA solution for $\Omega_R/\omega_0 = 0.01$, $0.1$, and $0.5$. (c) At what value of $\Omega_R/\omega_0$ does the RWA break down (say, when the maximum transition probability differs by more than 5% from the RWA prediction)?

Problem 7.27 ★★★ Ramsey interferometry. Two $\pi/2$-pulses are separated by a free evolution time $T$ (no driving field during $T$). (a) If the system starts in $|1\rangle$, what is the state after the first $\pi/2$-pulse? (b) After free evolution for time $T$, what relative phase has accumulated between $|1\rangle$ and $|2\rangle$? (c) After the second $\pi/2$-pulse, what is $P_2$ as a function of the detuning $\delta$ and the free evolution time $T$? (d) Explain why this technique is used in atomic clocks. What sets the frequency resolution?

Problem 7.28 ★★★★ Adiabatic passage. Instead of a sudden pulse, the driving frequency is slowly swept through resonance (a "chirped" pulse): $\omega(t) = \omega_0 + \alpha t$ where $\alpha$ is the sweep rate. (a) Argue qualitatively that if the sweep is slow enough, the system will be transferred from $|1\rangle$ to $|2\rangle$ with high probability regardless of the exact value of $\Omega_R$. (This is the adiabatic theorem — to be treated fully in Chapter 32.) (b) The Landau-Zener formula gives the probability of not making the transition as $P_{\text{fail}} = e^{-\pi\Omega_R^2/(2|\alpha|)}$. For $\Omega_R/(2\pi) = 10$ kHz, what sweep rate $\alpha$ gives 99% transfer probability? (c) Why is adiabatic passage more robust than a $\pi$-pulse in practice?