Chapter 10 Quiz

Instructions: Select the best answer for each question. Answers and explanations follow at the end.

Q1. In quantum mechanics, a symmetry transformation must be implemented by:

(a) A Hermitian operator (b) A unitary or antiunitary operator (c) Any linear operator (d) A projection operator

Q2. The generator of spatial translations is:

(a) The position operator $\hat{x}$ (b) The Hamiltonian $\hat{H}$ (c) The momentum operator $\hat{p}$ (d) The angular momentum operator $\hat{L}$

Q3. The quantum Noether theorem states that if $[\hat{H}, \hat{G}] = 0$, then:

(a) $\hat{G}$ and $\hat{H}$ have the same eigenvalues (b) $\hat{G}$ is the Hamiltonian of the system (c) $\langle\hat{G}\rangle$ is conserved in time (d) $\hat{G}$ must be the identity operator

Q4. Which of the following is the correct form of the finite translation operator?

(a) $\hat{T}(a) = e^{i\hat{x}a/\hbar}$ (b) $\hat{T}(a) = e^{-i\hat{p}a/\hbar}$ (c) $\hat{T}(a) = e^{-i\hat{H}a/\hbar}$ (d) $\hat{T}(a) = 1 - i\hat{p}a/\hbar$

Q5. A free particle has Hamiltonian $\hat{H} = \hat{p}^2/2m$. Which quantity is conserved?

(a) Position (b) Linear momentum (c) Angular momentum (d) Both (b) and (c)

Q6. The generator of rotations about the $z$-axis is:

(a) $\hat{L}_x$ (b) $\hat{L}_y$ (c) $\hat{L}_z$ (d) $\hat{L}^2$

Q7. Rotations about different axes do not commute. This is reflected in which algebraic relation?

(a) $[\hat{L}_i, \hat{L}_j] = 0$ (b) $[\hat{L}_i, \hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k$ (c) $\hat{L}_i\hat{L}_j = \hat{L}_j\hat{L}_i$ (d) $[\hat{L}_i, \hat{H}] = i\hbar\hat{L}_i$

Q8. For a central potential $V(r)$, which quantum numbers label the energy eigenstates due to rotational symmetry?

(a) Only $n$ (b) $n$ and $l$ (c) $n$, $l$, and $m$ (d) Only $l$ and $m$

Q9. The $(2l+1)$-fold degeneracy of hydrogen atom states with the same $n$ and $l$ but different $m$ is a consequence of:

(a) Time-reversal symmetry (b) Parity symmetry (c) Rotational symmetry (d) Translational symmetry

Q10. The eigenvalues of the parity operator $\hat{\Pi}$ are:

(a) $0$ and $1$ (b) $+1$ and $-1$ (c) Any real number (d) $+i$ and $-i$

Q11. Under parity, the momentum operator transforms as $\hat{\Pi}\hat{p}\hat{\Pi}^\dagger = $:

(a) $+\hat{p}$ (b) $-\hat{p}$ (c) $\hat{p}^2$ (d) $0$

Q12. Angular momentum $\hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}}$ is:

(a) Odd under parity (changes sign) (b) Even under parity (does not change sign) (c) Neither even nor odd (d) Undefined under parity

Q13. The parity selection rule for electric dipole transitions requires:

(a) $\Delta l = 0$ (b) $\Delta l = \pm 2$ (c) $\Delta l = \pm 1$ (d) $\Delta l$ can be any integer

Q14. For the QHO, the parity of the $n$-th eigenstate is:

(a) Always $+1$ (b) Always $-1$ (c) $(-1)^n$ (d) $(-1)^{n+1}$

Q15. Time reversal is implemented by an antiunitary operator because:

(a) Time is not a Hermitian operator (b) A unitary time reversal would give forward evolution of the reversed state, not backward evolution (c) The Schrodinger equation is second-order in time (d) The Hamiltonian is not bounded below

Q16. Under time reversal, the spin operator transforms as $\hat{\Theta}\hat{S}\hat{\Theta}^{-1} = $:

(a) $+\hat{S}$ (b) $-\hat{S}$ (c) $\hat{S}^2$ (d) $0$

Q17. Kramers' theorem states that for a system with time-reversal symmetry and half-integer total spin:

(a) All energy levels are non-degenerate (b) Every energy level is at least two-fold degenerate (c) The ground state energy is zero (d) Parity is conserved

Q18. Bloch's theorem applies to particles in:

(a) A constant potential (b) A random potential (c) A periodic potential (d) A central potential

Q19. In Bloch's theorem, the wave function takes the form $\psi_{nk}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{nk}(\mathbf{r})$, where $u_{nk}$:

(a) Is a plane wave (b) Has the periodicity of the crystal lattice (c) Decays exponentially (d) Is independent of position

Q20. A student claims: "If the Hamiltonian commutes with $\hat{G}$, then $\hat{H}$ and $\hat{G}$ must have the same eigenvalues." This statement is:

(a) True — commuting operators always share eigenvalues (b) True — this follows from the spectral theorem (c) False — they share eigenstates but have independent eigenvalues (d) False — commuting operators cannot share eigenstates

Answers and Explanations

Q1. Answer: (b)

Wigner's theorem establishes that any symmetry transformation — any map that preserves transition probabilities $|\langle\phi|\psi\rangle|^2$ — must be implemented by either a unitary operator (linear, preserving inner products) or an antiunitary operator (antilinear, conjugating inner products). Most symmetries (translation, rotation, parity) are unitary. Time reversal is the important exception: it is antiunitary.

Q2. Answer: (c)

The translation operator is $\hat{T}(a) = e^{-i\hat{p}a/\hbar}$. Comparing with the general form $\hat{U}(\alpha) = e^{-i\alpha\hat{G}/\hbar}$, we identify the generator as $\hat{G} = \hat{p}$. Physically: momentum is the "charge" of translational symmetry.

Q3. Answer: (c)

The quantum Noether theorem establishes the equivalence: $[\hat{H}, \hat{G}] = 0$ implies $\frac{d}{dt}\langle\hat{G}\rangle = 0$ (conservation). The operators share eigenstates but their eigenvalues are completely independent. $\hat{G}$ need not be the Hamiltonian, and it is not necessarily the identity.

Q4. Answer: (b)

The finite translation operator is obtained by exponentiating the generator: $\hat{T}(a) = e^{-i\hat{p}a/\hbar}$. Option (d) is only the first-order (infinitesimal) approximation.

Q5. Answer: (d)

The free particle Hamiltonian is invariant under both translation and rotation. $[\hat{H}, \hat{p}] = 0$ (momentum conserved) and $[\hat{H}, \hat{L}] = 0$ (angular momentum conserved, since $V = 0$ is trivially a central potential).

Q6. Answer: (c)

The rotation operator about $\hat{z}$ is $\hat{R}_z(\phi) = e^{-i\hat{L}_z\phi/\hbar}$, so the generator is $\hat{L}_z$. More generally, the generator of rotation about axis $\hat{n}$ is $\hat{\mathbf{J}} \cdot \hat{n}$.

Q7. Answer: (b)

The angular momentum commutation relations $[\hat{L}_i, \hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k$ encode the non-commutativity of rotations about different axes. This is the $\mathfrak{su}(2)$ Lie algebra.

Q8. Answer: (c)

Rotational symmetry allows simultaneous eigenstates of $\hat{H}$, $\hat{L}^2$, and $\hat{L}_z$, labeled by $n$ (from the radial equation), $l$ (from $\hat{L}^2$), and $m$ (from $\hat{L}_z$). The principal quantum number $n$ labels energy; $l$ and $m$ come from the symmetry group.

Q9. Answer: (c)

Rotational symmetry means $[\hat{H}, \hat{R}(\hat{n}, \theta)] = 0$. A rotation maps $|n, l, m\rangle$ to a linear combination of $|n, l, m'\rangle$ states with different $m$ but the same $l$ and $n$. Since $\hat{R}$ commutes with $\hat{H}$, all $2l+1$ states in the multiplet have the same energy.

Q10. Answer: (b)

Since $\hat{\Pi}^2 = \hat{I}$ (reflecting twice returns to the original), the eigenvalue $\pi$ satisfies $\pi^2 = 1$, giving $\pi = \pm 1$. States are either even ($+1$) or odd ($-1$) under parity.

Q11. Answer: (b)

Under parity, position reverses sign ($\hat{\Pi}\hat{x}\hat{\Pi}^\dagger = -\hat{x}$) and momentum reverses sign ($\hat{\Pi}\hat{p}\hat{\Pi}^\dagger = -\hat{p}$). Both are "polar vectors" — they are odd under parity.

Q12. Answer: (b)

$\hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}}$. Since both $\hat{\mathbf{r}}$ and $\hat{\mathbf{p}}$ are odd (each picks up a minus sign), their cross product is even: $(-1)(-1) = +1$. Angular momentum is a pseudovector (axial vector).

Q13. Answer: (c)

The electric dipole operator $\hat{d} = -e\hat{\mathbf{r}}$ is parity-odd. The matrix element $\langle n', l', m'|\hat{d}|n, l, m\rangle$ vanishes unless the initial and final states have opposite parity. Since parity of hydrogen states is $(-1)^l$, we need $(-1)^{l'} \neq (-1)^l$, i.e., $\Delta l = \pm 1$ (odd).

Q14. Answer: (c)

The QHO eigenstates have parity $(-1)^n$: even states ($n = 0, 2, 4, \ldots$) are symmetric, odd states ($n = 1, 3, 5, \ldots$) are antisymmetric. This follows from $H_n(-\xi) = (-1)^n H_n(\xi)$ for Hermite polynomials.

Q15. Answer: (b)

If time reversal were unitary (linear), applying it to the time evolution $|\psi(t)\rangle = e^{-i\hat{H}t/\hbar}|\psi(0)\rangle$ would give $e^{-i\hat{H}t/\hbar}\hat{\Theta}|\psi(0)\rangle$ — forward evolution of the reversed state. Antiunitarity gives $e^{+i\hat{H}t/\hbar}\hat{\Theta}|\psi(0)\rangle$ — backward evolution, which is what time reversal requires.

Q16. Answer: (b)

Time reversal reverses all angular momenta, including spin: $\hat{\Theta}\hat{S}\hat{\Theta}^{-1} = -\hat{S}$. Physically, reversing the arrow of time reverses all rotations and angular velocities.

Q17. Answer: (b)

Kramers' theorem: for half-integer spin, $\hat{\Theta}^2 = -\hat{I}$, which implies $\langle\psi|\hat{\Theta}\psi\rangle = 0$. The state $|\psi\rangle$ and its time-reverse $\hat{\Theta}|\psi\rangle$ are orthogonal and degenerate. This two-fold degeneracy is broken only by a magnetic field (which breaks time-reversal symmetry).

Q18. Answer: (c)

Bloch's theorem applies to systems with discrete translational symmetry — periodic potentials $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$ for lattice vectors $\mathbf{R}$. This is the symmetry of a crystal lattice.

Q19. Answer: (b)

In $\psi_{nk}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{nk}(\mathbf{r})$, the function $u_{nk}$ has the periodicity of the lattice: $u_{nk}(\mathbf{r} + \mathbf{R}) = u_{nk}(\mathbf{r})$. The Bloch wave is a plane wave modulated by a lattice-periodic function.

Q20. Answer: (c)

Commuting Hermitian operators can be simultaneously diagonalized — they share a complete set of eigenstates. But their eigenvalues are independent. For example, $[\hat{H}, \hat{L}^2] = 0$ for the hydrogen atom, but $\hat{H}$ has eigenvalues $-13.6/n^2$ eV and $\hat{L}^2$ has eigenvalues $\hbar^2 l(l+1)$. Same eigenstates, different spectra.