Chapter 9 Quiz

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

Q1. The eigenvalue equation $\hat{A}|a\rangle = a|a\rangle$ implies that:

(a) $\hat{A}$ is always invertible (b) $|a\rangle$ is unchanged (up to a scalar multiple) when $\hat{A}$ acts on it (c) $a$ must be a positive real number (d) $|a\rangle$ is the only state in the Hilbert space with this property

Q2. For a Hermitian operator $\hat{A}$, the eigenvalues are guaranteed to be:

(a) Positive (b) Integer (c) Real (d) Non-degenerate

Q3. The eigenstates of $\hat{S}_x$ for a spin-1/2 particle, expressed in the $\hat{S}_z$ eigenbasis, are:

(a) $|\uparrow\rangle$ and $|\downarrow\rangle$ (b) $\frac{1}{\sqrt{2}}(|\uparrow\rangle \pm |\downarrow\rangle)$ (c) $\frac{1}{\sqrt{2}}(|\uparrow\rangle \pm i|\downarrow\rangle)$ (d) $\frac{1}{2}(|\uparrow\rangle + \sqrt{3}|\downarrow\rangle)$ and $\frac{1}{2}(\sqrt{3}|\uparrow\rangle - |\downarrow\rangle)$

Q4. The normalization condition for position eigenstates is:

(a) $\langle x|x'\rangle = 1$ if $x = x'$, $0$ otherwise (b) $\langle x|x'\rangle = \delta_{xx'}$ (Kronecker delta) (c) $\langle x|x'\rangle = \delta(x - x')$ (Dirac delta) (d) $\langle x|x'\rangle = \frac{1}{2\pi\hbar}e^{i(x-x')/\hbar}$

Q5. The wave function $\psi(x) = \langle x|\psi\rangle$ is best described as:

(a) The quantum state itself (b) The probability of finding the particle at position $x$ (c) The expansion coefficient of $|\psi\rangle$ in the position eigenbasis (d) The eigenvalue of the position operator

Q6. The Dirac delta function $\delta(x - a)$ is defined by:

(a) $\delta(x - a) = \infty$ if $x = a$, $0$ otherwise (b) $\int_{-\infty}^{\infty} f(x)\delta(x - a) \, dx = f(a)$ for any continuous $f$ (c) $\delta(x - a) = \frac{d}{dx}\theta(x - a)$ where $\theta$ is the step function (d) Both (b) and (c) are correct definitions

Q7. The scaling property of the delta function gives $\delta(3x) = $:

(a) $3\delta(x)$ (b) $\frac{1}{3}\delta(x)$ (c) $\delta(x)$ (d) $9\delta(x)$

Q8. The Fourier integral representation of the delta function is:

(a) $\delta(x) = \int_{-\infty}^{\infty} e^{ikx} \, dk$ (b) $\delta(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx} \, dk$ (c) $\delta(x) = \frac{1}{(2\pi)^2}\int_{-\infty}^{\infty} e^{ikx} \, dk$ (d) $\delta(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{ikx} \, dk$

Q9. The spectral decomposition of a Hermitian operator $\hat{A}$ with eigenvalues $\{a_n\}$ and eigenstates $\{|a_n\rangle\}$ is:

(a) $\hat{A} = \sum_n |a_n\rangle\langle a_n|$ (b) $\hat{A} = \sum_n a_n^2 |a_n\rangle\langle a_n|$ (c) $\hat{A} = \sum_n a_n |a_n\rangle\langle a_n|$ (d) $\hat{A} = \sum_n \frac{1}{a_n} |a_n\rangle\langle a_n|$

Q10. Using the spectral decomposition, the time-evolution operator $e^{-i\hat{H}t/\hbar}$ equals:

(a) $\sum_n e^{-iE_n t/\hbar}|E_n\rangle$ (b) $\sum_n (-iE_n t/\hbar)^n|E_n\rangle\langle E_n|$ (c) $\sum_n e^{-iE_n t/\hbar}|E_n\rangle\langle E_n|$ (d) $e^{-it/\hbar}\sum_n E_n|E_n\rangle\langle E_n|$

Q11. For a spin-1/2 particle, $e^{-i\hat{S}_z \cdot 2\pi/\hbar}$ equals:

(a) $\hat{I}$ (the identity) (b) $-\hat{I}$ (minus the identity) (c) $\hat{S}_z$ (d) $0$

Q12. The Fourier transform from position to momentum representation is:

(a) $\phi(p) = \int \psi(x) \, dx$ (b) $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \psi(x) e^{ipx/\hbar} \, dx$ (c) $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \psi(x) e^{-ipx/\hbar} \, dx$ (d) $\phi(p) = \frac{1}{2\pi\hbar}\int \psi(x) e^{-ipx/\hbar} \, dx$

Q13. The Fourier transform of a Gaussian wave function is:

(a) A Lorentzian (b) A Gaussian (c) A sinc function (d) A delta function

Q14. Parseval's theorem states that:

(a) $\int|\psi(x)|^2 dx = 1$ (b) $\int|\psi(x)|^2 dx = \int|\phi(p)|^2 dp$ (c) $\int\psi(x)\phi(p) \, dx = \delta(p)$ (d) $|\psi(x)|^2 = |\phi(p)|^2$ for all $x$ and $p$

Q15. In the momentum representation, the position operator $\hat{x}$ acts as:

(a) Multiplication by $x$ (b) $-i\hbar\frac{\partial}{\partial p}$ (c) $i\hbar\frac{\partial}{\partial p}$ (d) $\frac{p}{m}$

Q16. A position eigenstate $|x_0\rangle$ has momentum-space wave function:

(a) $\phi(p) = \delta(p - p_0)$ (b) $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}e^{-ipx_0/\hbar}$ (a plane wave) (c) $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}e^{+ipx_0/\hbar}$ (d) $\phi(p) = 0$

Q17. The rigged Hilbert space $\Phi \subset \mathcal{H} \subset \Phi'$ is needed because:

(a) The Hilbert space is too small — it does not contain generalized eigenstates like $|x\rangle$ (b) The Hilbert space is too large — it contains unphysical states (c) Dirac notation only works in the rigged Hilbert space (d) The spectral theorem fails in the ordinary Hilbert space

Q18. In the rigged Hilbert space, physically preparable states belong to:

(a) $\Phi'$ (the dual space) (b) $\mathcal{H}$ (the Hilbert space) but not $\Phi$ (c) $\Phi$ (the Schwartz space) (d) The quotient space $\Phi'/\mathcal{H}$

Q19. A quantum system has Hamiltonian $\hat{H}$ with both bound states (energies $E_1, E_2, \ldots$) and scattering states (energies $E > 0$). The completeness relation is:

(a) $\sum_n |E_n\rangle\langle E_n| = \hat{I}$ (b) $\int_0^\infty |E\rangle\langle E| \, dE = \hat{I}$ (c) $\sum_n |E_n\rangle\langle E_n| + \int_0^\infty |E\rangle\langle E| \, dE = \hat{I}$ (d) $\sum_n |E_n\rangle\langle E_n| \cdot \int_0^\infty |E\rangle\langle E| \, dE = \hat{I}$

Q20. The Gaussian wave function $\psi(x) \propto e^{-x^2/4\sigma^2}$ achieves $\Delta x \Delta p = \hbar/2$. This means the Gaussian is:

(a) The only normalizable state (b) The minimum-uncertainty state (c) The ground state of any quantum system (d) The eigenstate of both $\hat{x}$ and $\hat{p}$

Answers and Explanations

Q1: (b) The eigenvalue equation says $\hat{A}$ maps $|a\rangle$ to $a|a\rangle$ — the "direction" in Hilbert space is unchanged, only the "length" (scalar factor) changes. Eigenvalues can be negative (ruling out (c)), the eigenvalue can be degenerate (ruling out (d)), and singular operators have eigenvalue 0 (ruling out (a)).

Q2: (c) Hermiticity $\hat{A} = \hat{A}^\dagger$ guarantees real eigenvalues. The proof (Section 9.1): $\langle a|\hat{A}|a\rangle = a\langle a|a\rangle = a$, and also $\langle a|\hat{A}|a\rangle = \langle a|\hat{A}^\dagger|a\rangle^* = a^*$, so $a = a^*$, meaning $a$ is real. Eigenvalues need not be positive (e.g., $\hat{S}_z$ has eigenvalue $-\hbar/2$), integer, or non-degenerate.

Q3: (b) $|+\rangle_x = \frac{1}{\sqrt{2}}(|\uparrow\rangle + |\downarrow\rangle)$ and $|-\rangle_x = \frac{1}{\sqrt{2}}(|\uparrow\rangle - |\downarrow\rangle)$. Option (c) gives the $\hat{S}_y$ eigenstates.

Q4: (c) Position eigenvalues form a continuum, so orthonormality uses the Dirac delta rather than the Kronecker delta. This is the hallmark of continuous-spectrum operators.

Q5: (c) The wave function is the coefficient obtained by expanding $|\psi\rangle$ in the position basis: $|\psi\rangle = \int \psi(x)|x\rangle dx$ with $\psi(x) = \langle x|\psi\rangle$. It is not the state itself (that is $|\psi\rangle$), not the probability (that is $|\psi(x)|^2$), and not an eigenvalue.

Q6: (d) Both (b) and (c) are valid definitions. The sifting property (b) is the standard definition in distribution theory. The derivative of the Heaviside step function (c) is an equivalent characterization. Option (a) is an informal description that, taken literally, does not define a distribution.

Q7: (b) The scaling property gives $\delta(ax) = \frac{1}{|a|}\delta(x)$. With $a = 3$: $\delta(3x) = \frac{1}{3}\delta(x)$.

Q8: (b) The correct Fourier integral representation is $\delta(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx} dk$. The factor of $1/(2\pi)$ comes from the symmetric Fourier transform convention.

Q9: (c) The spectral decomposition weights each projector $|a_n\rangle\langle a_n|$ by the corresponding eigenvalue $a_n$. Option (a) is the completeness relation (identity operator), not the spectral decomposition.

Q10: (c) By the spectral theorem, $f(\hat{H}) = \sum_n f(E_n)|E_n\rangle\langle E_n|$ with $f(E) = e^{-iEt/\hbar}$. Both the ket and the bra are needed — this is a sum of projectors, not a sum of kets.

Q11: (b) $e^{-i\hat{S}_z \cdot 2\pi/\hbar} = e^{-i\pi}|\uparrow\rangle\langle\uparrow| + e^{i\pi}|\downarrow\rangle\langle\downarrow| = (-1)|\uparrow\rangle\langle\uparrow| + (-1)|\downarrow\rangle\langle\downarrow| = -\hat{I}$. A spin-1/2 particle picks up a sign under $2\pi$ rotation.

Q12: (c) The convention with the minus sign in the exponent takes position to momentum: $\phi(p) = (2\pi\hbar)^{-1/2}\int \psi(x)e^{-ipx/\hbar}dx$. The inverse (momentum to position) has the plus sign.

Q13: (b) The Fourier transform of a Gaussian is a Gaussian. A narrow Gaussian in position transforms to a wide Gaussian in momentum, and vice versa, consistent with the uncertainty principle.

Q14: (b) Parseval's theorem says the total probability is the same in both representations: $\int|\psi(x)|^2 dx = \int|\phi(p)|^2 dp = \langle\psi|\psi\rangle$. This follows from the completeness of both the position and momentum bases.

Q15: (c) In momentum representation, $\hat{x}$ acts as $i\hbar\frac{\partial}{\partial p}$, derived by inserting a complete set of position states (Section 9.6). Note the sign: position in momentum representation has a plus sign, while momentum in position representation has a minus sign: $\hat{p} \to -i\hbar\frac{\partial}{\partial x}$.

Q16: (b) $\phi(p) = \langle p|x_0\rangle = (2\pi\hbar)^{-1/2}e^{-ipx_0/\hbar}$, a plane wave with constant amplitude $1/\sqrt{2\pi\hbar}$. Every momentum is equally probable — perfect position knowledge implies total momentum ignorance.

Q17: (a) The Hilbert space $L^2(\mathbb{R})$ contains only square-integrable functions. Position eigenstates $|x\rangle$ and momentum eigenstates $|p\rangle$ have infinite norm and are not in $L^2$. The rigged Hilbert space extends $\mathcal{H}$ to the dual space $\Phi'$, which accommodates these generalized eigenstates.

Q18: (c) Physical states are smooth, rapidly decreasing functions — elements of the Schwartz space $\Phi$, the "inner box" of the Gelfand triple. Generalized eigenstates live in $\Phi'$ (the "outer box"), and $\mathcal{H}$ is the closure of $\Phi$.

Q19: (c) A system with a mixed spectrum has both discrete bound-state eigenstates and continuous scattering-state eigenstates. Both contribute to the completeness relation. Neither the sum alone nor the integral alone is complete.

Q20: (b) The Gaussian saturates the Heisenberg uncertainty relation $\Delta x \Delta p \geq \hbar/2$, achieving equality. This makes it the minimum-uncertainty state. It is not an eigenstate of $\hat{x}$ or $\hat{p}$ (those have infinite uncertainty in the conjugate variable), and it is the ground state only of the QHO, not of general systems.