Chapter 7 Quiz

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 7 Quiz

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

Q1. The time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ is guaranteed to be unitary because:

(a) The Hamiltonian is always positive definite (b) The Hamiltonian is Hermitian (c) The Hamiltonian commutes with itself at different times (d) The exponential function always produces unitary operators

Q2. A particle is in the energy eigenstate $\psi_n$ of some time-independent Hamiltonian. Which of the following is time-dependent?

(a) The probability density $|\Psi_n(x,t)|^2$ (b) The expectation value $\langle\hat{x}\rangle$ (c) The phase of the wave function (d) The expectation value $\langle\hat{H}\rangle$

Q3. A particle in an infinite square well is in the state $\Psi(x,0) = (1/\sqrt{2})[\psi_1(x) + \psi_2(x)]$. The probability density $|\Psi(x,t)|^2$ oscillates with angular frequency:

(a) $E_1/\hbar$ (b) $E_2/\hbar$ (c) $(E_2 - E_1)/\hbar$ (d) $(E_2 + E_1)/\hbar$

Q4. The group velocity of a free-particle wave packet with average wave number $k_0$ is:

(a) $\hbar k_0/(2m)$ (half the phase velocity) (b) $\hbar k_0/m$ (the classical velocity $p_0/m$) (c) $2\hbar k_0/m$ (twice the phase velocity) (d) $\omega_0/k_0$ (the phase velocity)

Q5. A Gaussian wave packet for a free electron has initial width $\sigma_0$. After a long time ($t \gg 2m\sigma_0^2/\hbar$), the width grows approximately as:

(a) $\sigma(t) \propto t^{1/2}$ (b) $\sigma(t) \propto t$ (c) $\sigma(t) \propto t^2$ (d) $\sigma(t)$ approaches a constant (no further spreading)

Q6. The revival time for a particle in an infinite square well of width $a$ is $T_{\text{rev}} = 4ma^2/(\pi\hbar)$. If the well width is doubled, the revival time:

(a) Doubles (b) Quadruples (c) Halves (d) Stays the same

Q7. At time $t = T_{\text{rev}}/2$ in an infinite square well, a wave packet initially localized near $x = a/4$ will be found:

(a) Still localized near $x = a/4$ (b) Localized near $x = 3a/4$ (mirror image position) (c) Spread uniformly across the well (d) Split into two equal copies

Q8. In the Schrödinger picture:

(a) States are fixed, operators evolve (b) States evolve, operators are fixed (c) Both states and operators evolve (d) Neither states nor operators evolve

Q9. The Heisenberg equation of motion $d\hat{A}_H/dt = (i/\hbar)[\hat{H}, \hat{A}_H]$ is the quantum analogue of:

(a) Newton's second law (b) The wave equation (c) Hamilton's equations via Poisson brackets (d) The continuity equation

Q10. If an operator $\hat{A}$ commutes with $\hat{H}$, then in the Heisenberg picture:

(a) $\hat{A}_H(t)$ oscillates sinusoidally (b) $\hat{A}_H(t) = \hat{A}_H(0)$ — the operator is constant (c) $\hat{A}_H(t)$ grows linearly in time (d) The behavior depends on the specific form of $\hat{H}$

Q11. In the interaction picture with $\hat{H} = \hat{H}_0 + \hat{V}$, the operators evolve according to:

(a) The full Hamiltonian $\hat{H}$ (b) The perturbation $\hat{V}$ only (c) The unperturbed Hamiltonian $\hat{H}_0$ only (d) Neither $\hat{H}_0$ nor $\hat{V}$

Q12. Ehrenfest's theorem states that $d\langle\hat{p}\rangle/dt = -\langle dV/dx\rangle$. This exactly reproduces Newton's second law when:

(a) The particle is in an energy eigenstate (b) The potential is at most quadratic in $x$ (e.g., harmonic oscillator) (c) $\hbar \to 0$ (d) The particle has zero average momentum

Q13. For the quantum harmonic oscillator, Ehrenfest's theorem gives exact classical equations for $\langle\hat{x}\rangle$ and $\langle\hat{p}\rangle$ regardless of the state. This is because:

(a) The QHO has equally spaced energy levels (b) The force $F = -m\omega^2 x$ is linear in $x$, so $\langle F(\hat{x})\rangle = F(\langle\hat{x}\rangle)$ (c) The QHO ground state is a Gaussian (d) The QHO commutator $[\hat{a}, \hat{a}^\dagger] = 1$ is a c-number

Q14. In a Rabi oscillation experiment at exact resonance ($\delta = 0$), the system oscillates between states $|1\rangle$ and $|2\rangle$. The population of $|2\rangle$ first reaches 100% at time:

(a) $t = \pi/(2\Omega_R)$ ($\pi/2$-pulse) (b) $t = \pi/\Omega_R$ ($\pi$-pulse) (c) $t = 2\pi/\Omega_R$ ($2\pi$-pulse) (d) $t = 1/\Omega_R$

Q15. If the driving frequency is detuned from resonance ($\delta \neq 0$), the maximum transition probability $P_{1\to 2}^{\max}$ is:

(a) Always 1 (complete population transfer still occurs) (b) $\Omega_R^2/(\Omega_R^2 + \delta^2) < 1$ (c) $\delta^2/(\Omega_R^2 + \delta^2) < 1$ (d) $\Omega_R/\delta$

Q16. The rotating wave approximation (RWA) in the driven two-level problem is justified when:

(a) The driving field is weak ($\Omega_R \ll \omega_0$) (b) The detuning is small ($\delta \ll \omega_0$) (c) Both (a) and (b) (d) The driving field is on resonance ($\delta = 0$)

Q17. A $\pi/2$-pulse applied to a qubit initially in state $|0\rangle$ produces:

Q18. The time-ordered exponential $\mathcal{T}\exp(...)$ is needed for the time-evolution operator when:

(a) The Hamiltonian is time-independent (b) The Hamiltonian at different times does not commute: $[\hat{H}(t_1), \hat{H}(t_2)] \neq 0$ (c) The system has more than two energy levels (d) The potential is unbounded

Q19. Consider a quantum system with Hamiltonian $\hat{H} = \hat{p}^2/(2m)$ (free particle). In the Heisenberg picture, $\hat{p}_H(t)$ is:

(a) $\hat{p}(0)\cos(\omega t)$ (b) $\hat{p}(0) + \hat{F}t$ (c) $\hat{p}(0)$ (constant in time) (d) $\hat{p}(0)e^{-i\omega t}$

Q20. Which of the following best describes why all three pictures (Schrödinger, Heisenberg, interaction) give the same physics?

(a) They use the same Hamiltonian (b) They are related by unitary transformations, which preserve all inner products and expectation values (c) They all use the same wave function (d) The correspondence principle guarantees their equivalence