Chapter 6 Quiz

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 6 Quiz

Instructions: Select the best answer for each question. Each question has exactly one correct answer unless stated otherwise.

Question 1

What distinguishes a linear operator from a nonlinear one?

(A) A linear operator always produces real outputs.

(B) A linear operator satisfies $\hat{A}(\alpha\psi + \beta\phi) = \alpha\hat{A}\psi + \beta\hat{A}\phi$.

(C) A linear operator commutes with all other operators.

(D) A linear operator has a finite number of eigenvalues.

Question 2

Which of the following is the correct position-space representation of the kinetic energy operator $\hat{T}$?

(A) $\hat{T} = -i\hbar\frac{d}{dx}$

(B) $\hat{T} = \frac{1}{2}m\hat{v}^2$

(C) $\hat{T} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$

(D) $\hat{T} = \frac{\hat{p}}{m}$

Question 3

Why must quantum mechanical observables correspond to Hermitian operators?

(A) Because Hermitian operators are easier to compute with.

(B) Because Hermitian operators have real eigenvalues, and measurement results must be real numbers.

(C) Because only Hermitian operators are linear.

(D) Because Hermitian operators always commute with the Hamiltonian.

Question 4

The commutator $[\hat{x}, \hat{p}]$ equals:

(A) $0$

(B) $\hbar$

(C) $i\hbar$

(D) $-i\hbar$

Question 5

If two Hermitian operators $\hat{A}$ and $\hat{B}$ commute ($[\hat{A}, \hat{B}] = 0$), which statement is TRUE?

(A) They cannot be measured simultaneously.

(B) They share no common eigenstates.

(C) They can be simultaneously diagonalized — they share a complete set of common eigenstates.

(D) Their uncertainty product is always $\hbar/2$.

Question 6

The commutator $[\hat{x}, \hat{p}^2]$ equals:

(A) $i\hbar$

(B) $2i\hbar\hat{p}$

(C) $i\hbar\hat{p}^2$

(D) $2i\hbar\hat{x}$

Question 7

Which of the following is the correct statement of the generalized uncertainty principle?

(A) $\sigma_A \sigma_B = \frac{1}{2}|\langle [\hat{A}, \hat{B}] \rangle|$

(B) $\sigma_A \sigma_B \geq \frac{1}{2}|\langle [\hat{A}, \hat{B}] \rangle|$

(C) $\sigma_A + \sigma_B \geq \frac{1}{2}|\langle [\hat{A}, \hat{B}] \rangle|$

(D) $\sigma_A \sigma_B \geq |\langle [\hat{A}, \hat{B}] \rangle|^2$

Question 8

The Heisenberg uncertainty principle $\Delta x \, \Delta p \geq \hbar/2$ is saturated (equality holds) for:

(A) Any energy eigenstate.

(B) The ground state of the infinite square well.

(C) Gaussian wave packets.

(D) Plane waves $e^{ikx}$.

Question 9

In the commutator identity $[\hat{A}, \hat{B}\hat{C}] = [\hat{A}, \hat{B}]\hat{C} + \hat{B}[\hat{A}, \hat{C}]$, the structure is analogous to:

(A) The chain rule of calculus.

(B) The product (Leibniz) rule of differentiation.

(C) Integration by parts.

(D) The triangle inequality.

Question 10

For the QHO ladder operators, $[\hat{a}, \hat{a}^\dagger]$ equals:

(A) $0$

(B) $1$

(C) $i\hbar$

(D) $\hat{a}^\dagger\hat{a}$

Question 11

The energy-time uncertainty relation $\Delta E \Delta t \geq \hbar/2$ differs from $\Delta x \Delta p \geq \hbar/2$ because:

(A) Energy is not an observable in quantum mechanics.

(B) Time is not an observable — there is no Hermitian operator $\hat{t}$.

(C) The energy-time relation is only approximate.

(D) The energy-time relation applies only to relativistic systems.

Question 12

An excited atomic state has a lifetime $\tau = 10^{-8}$ s. The natural linewidth of the emitted photon is approximately:

(A) $\Delta\nu \sim 10^{-8}$ Hz

(B) $\Delta\nu \sim 10^{7}$ Hz

(C) $\Delta\nu \sim 10^{15}$ Hz

(D) $\Delta\nu \sim 10^{20}$ Hz

Question 13

A complete set of commuting observables (CSCO) for the hydrogen atom (without spin) is:

(A) $\{\hat{H}\}$

(B) $\{\hat{H}, \hat{L}_z\}$

(C) $\{\hat{H}, \hat{L}^2, \hat{L}_z\}$

(D) $\{\hat{H}, \hat{L}^2, \hat{L}_z, \hat{L}_x\}$

Question 14

After measuring observable $\hat{A}$ and obtaining eigenvalue $a_n$, the state of the system is:

(A) Unchanged — measurement reveals pre-existing values.

(B) A superposition of all eigenstates with nonzero overlap.

(C) The eigenstate $\psi_n$ corresponding to eigenvalue $a_n$.

(D) The ground state of $\hat{A}$.

Question 15

In the Stern-Gerlach experiment, a beam of spin-1/2 particles prepared with $S_z = +\hbar/2$ is sent through an $x$-oriented analyzer. The probability of measuring $S_x = +\hbar/2$ is:

(A) 0

(B) 1/4

(C) 1/2

(D) 1

Question 16

If $\hat{A}$ is Hermitian with eigenstates $\psi_1, \psi_2$ corresponding to eigenvalues $a_1 \neq a_2$, then $\langle \psi_1 | \psi_2 \rangle$ equals:

(A) $1$

(B) $a_1 a_2$

(C) $0$

(D) $(a_1 - a_2)^{-1}$

Question 17

The expectation value $\langle \hat{A} \rangle$ represents:

(A) The most probable measurement outcome.

(B) The result of a single measurement.

(C) The average over many identical measurements on identically prepared systems.

(D) The eigenvalue closest to the mean energy.

Question 18

Which commutator identity is FALSE?

(A) $[\hat{A}, \hat{B}] = -[\hat{B}, \hat{A}]$

(B) $[\hat{A}, \hat{A}] = 0$

(C) $[\hat{A}, \hat{B} + \hat{C}] = [\hat{A}, \hat{B}] + [\hat{A}, \hat{C}]$

(D) $[\hat{A}, \hat{B}\hat{C}] = [\hat{A}, \hat{B}][\hat{A}, \hat{C}]$

Question 19

The measurement problem in quantum mechanics refers to:

(A) The difficulty of measuring very small quantities precisely.

(B) The tension between the linear, deterministic Schrodinger equation and the nonlinear, probabilistic collapse postulate.

(C) The impossibility of measuring two observables simultaneously.

(D) The finite precision of all laboratory instruments.

Question 20

For two observables with $[\hat{A}, \hat{B}] = i\hbar\hat{C}$ where $\hat{C}$ is Hermitian, the generalized uncertainty principle gives:

(A) $\sigma_A\sigma_B \geq \frac{\hbar}{2}|\langle \hat{C} \rangle|$

(B) $\sigma_A\sigma_B \geq \hbar$

(C) $\sigma_A\sigma_B \geq \frac{1}{2}|\langle \hat{C} \rangle|$

(D) $\sigma_A\sigma_B \geq \frac{\hbar^2}{4}\langle \hat{C}^2 \rangle$

Answer Key

(B) — The definition of linearity.
(C) — $\hat{T} = \hat{p}^2/(2m) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$.
(B) — Real eigenvalues guarantee real measurement outcomes.
(C) — The canonical commutation relation.
(C) — The simultaneous diagonalizability theorem (Theorem 6.3).
(B) — Calculated using the product rule: $[\hat{x}, \hat{p}^2] = [\hat{x},\hat{p}]\hat{p} + \hat{p}[\hat{x},\hat{p}] = 2i\hbar\hat{p}$.
(B) — The Robertson uncertainty relation (Eq. 6.4).
(C) — Gaussian wave packets are the minimum uncertainty states.
(B) — The structure mirrors the Leibniz product rule for derivatives.
(B) — Derived in Worked Example 6.5.
(B) — Time is a parameter in QM, not an observable with a Hermitian operator.
(B) — $\Delta\nu \sim 1/(2\pi\tau) \sim 1/(2\pi \times 10^{-8}) \sim 1.6 \times 10^7$ Hz.
(C) — $\hat{H}$ gives $n$, $\hat{L}^2$ gives $l$, $\hat{L}_z$ gives $m$; together they uniquely label states.
(C) — The projection postulate (state collapse).
(C) — The $S_z$ eigenstate is an equal superposition of $S_x$ eigenstates.
(C) — Theorem 6.2: eigenstates of a Hermitian operator with different eigenvalues are orthogonal.
(C) — The expectation value is a statistical average, not a single-shot result.
(D) — The correct identity is $[\hat{A}, \hat{B}\hat{C}] = [\hat{A}, \hat{B}]\hat{C} + \hat{B}[\hat{A}, \hat{C}]$ (not a product of commutators).
(B) — The measurement problem is about the conceptual tension between unitary evolution and collapse.
(A) — $\sigma_A\sigma_B \geq \frac{1}{2}|\langle [\hat{A},\hat{B}] \rangle| = \frac{1}{2}|i\hbar\langle\hat{C}\rangle| = \frac{\hbar}{2}|\langle\hat{C}\rangle|$.