8. Which of the following physical quantities has the same dimension as $\sqrt{\hbar(m\omega)^{-1}}$? (m: mass, $\omega$: angular frequency)
(A) momentum
(B) time
(C) energy
(D) speed
(E) distance

9. A particle of mass $m$ is subject to a one-dimensional attractive delta potential given by: $$ V(x) = -\alpha \delta(x), $$ where $\alpha > 0$. This potential supports exactly one bound state. Determine the energy eigenvalue $E$ and the normalized eigenfunction $\psi(x)$ for this state. We define $\kappa = \sqrt{-2mE} / \hbar$. (A) $E = -m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{-\kappa |x|}$ (B) $E = -m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{\kappa |x|}$ (C) $E = -m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{-\kappa x^2}$ (D) $E = +m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{-\kappa x^2}$ (E) $E = -\hbar^2 / 2m\alpha^2$ ; $\psi(x) = \sqrt{\kappa} e^{+\kappa |x|}$

10. In quantum mechanics, which of the following pairs of operators share simultaneous eigenstates? (A) Position along the $x$-axis $(\hat{x})$ and linear momentum along the $x$-axis $(\hat{p}_x)$. (B) The $x$-component of angular momentum $(\hat{L}_x)$ and the $y$-component of angular momentum $(\hat{L}_y)$. (C) Position along the $x$-axis $(\hat{x})$ and the $z$-component of angular momentum $(\hat{L}_z)$. (D) The squared magnitude of orbital angular momentum $(\hat{L}^2)$ and the $z$-component of orbital angular momentum $(\hat{L}_z)$. (E) The $x$-component of spin angular momentum $(\hat{S}_x)$ and the $z$-component of spin angular momentum $(\hat{S}_z)$. 

11. Consider a quantum particle in a one-dimensional infinite square well with potential $V(x) = 0$ for $-L < x < L$ and $V(x) = \infty$ for $|x| \geq L$. The particle has an initial wave function $\psi(x, t = 0) = \frac{1}{\sqrt{3}} \psi_1(x) + \sqrt{\frac{2}{3}} \psi_2(x)$, where $\psi_1$ and $\psi_2$ are the ground state and the first excited state of the system with eigenenergies $E_1$ and $E_2$, respectively. $\psi_{1,2}$ are real functions. What are the probability density $|\psi(x,t)|^2$ and the expectation value of energy $\langle H \rangle$ at later times $t > 0$? (A) $\langle H\rangle = \frac{1}{3} E_1 + \frac{2}{3} E_2$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2 +\frac{2\sqrt{2}}{3}\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$ (B) $\langle H\rangle \neq \frac{1}{2} (E_1 + E_2)$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2 +\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$ (C) $\langle H\rangle = \frac{1}{3} E_1 + \frac{2}{3} E_2$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2$ (D) $\langle H\rangle = \frac{1}{3} E_1 + \frac{2}{3} E_2\cos (t)$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2 +\frac{\sqrt{2}}{3}\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$ (E) $\langle H\rangle = E_1 + \sqrt{2} E_2$ ; $|\Psi (x,t)|^2 = \psi_1^2 +2\psi_2^2 +2\sqrt{2}\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$

12. The Pauli-X matrix, denoted by $\hat{\sigma}_x$, is represented as $$ \hat {\sigma} _ {x} = \left( \begin{array}{c c} 0 & 1 \\ 1 & 0 \end{array} \right). $$ Find the expectation value $\langle \hat{\sigma}_x\rangle$ in the spin state: $$ | \psi \rangle = \alpha_ {+} \left( \begin{array}{c} 1 \\ 0 \end{array} \right) + \alpha_ {-} \left( \begin{array}{c} 0 \\ 1 \end{array} \right), $$ where $\alpha_{+}$ and $\alpha_{-}$ are complex numbers. (A) 0 (B) $2\operatorname{Re}(\alpha_{+}^{*}\alpha_{-})$ (C) $2\operatorname{Im}(\alpha_{+}^{*}\alpha_{-})$ (D) $|\alpha_{+}|^{2} - |\alpha_{-}|^{2}$ (E) $|\alpha_{+}|^{2} + |\alpha_{-}|^{2}$

13. Consider a free electron gas in $d$ spatial dimensions. Let $n$ be the electron number density (defined as the number of electrons per unit length, area, or volume, depending on the dimension). Which of the following correctly describes the dependence of the Fermi energy $E_{F}$ on the density $n$? (A) $E_{F}\propto n^{2 / d}$ (B) $E_{F}\propto n^{d / 2}$ (C) $E_{F}\propto n$ (D) $E_{F}\propto n^{2}$ (E) $E_{F}\propto n^{1 / d}$

14. Consider a spinless particle in a three-dimensional isotropic simple harmonic oscillator potential. The particle is in an energy eigenstate with eigenenergy: $$ E = \left(99 + \frac{3}{2}\right) \hbar \omega $$ What is the degeneracy of this energy level? (A) 2500 (B) 5151 (C) 10100 (D) 100 (E) 5050

15. The total wavefunction of two electrons is written as a product of a spatial part $\psi(\vec{r}_1, \vec{r}_2)$ and a spin part $\chi_{spin}(s_1, s_2)$. Which of the following combinations correctly describes the symmetry of the spatial and spin parts specifically for the ground state of helium? (A) Spatial part is symmetric; Spin part is antisymmetric. (B) Spatial part is symmetric; Spin part is symmetric. (C) Spatial part is antisymmetric; Spin part is symmetric. (D) Spatial part is antisymmetric; Spin part is antisymmetric. (E) The Pauli principle forbids two electrons from occupying the same 1s spatial orbital, regardless of spin state.

16. Which of the following mathematical expressions correctly represents Planck's formula for the energy density per unit frequency $\rho(\omega)$ of a blackbody at absolute temperature $T$? ($k_B$: Boltzmann constant, $\omega$: angular frequency of light.) (A) $\frac{\hbar \omega^3}{\pi^2 c^3} \frac{1}{e^{\hbar \omega / (k_B T)} - 1}$ (B) $\frac{\hbar \omega^3}{\pi^2 c^3} k_B T$ (C) $\frac{\hbar \omega^3}{\pi^2 c^3} e^{-\hbar \omega / (k_B T)}$ (D) $\frac{\hbar \omega^3}{\pi^2 c^3} \frac{1}{e^{\hbar \omega / (k_B T)} + 1}$ (E) $\frac{\hbar \omega}{e^{\hbar \omega / (k_B T)} - 1}$

複選題17. A harmonic oscillator with charge $q$ is initially in an eigenstate $|n\rangle$. It is subjected to a time-dependent electric field $\mathcal{E}(t)$ along the $x$-axis, resulting in the perturbing Hamiltonian $\hat{H}'(t) = -q\mathcal{E}(t)\hat{x}$. Based on the properties of the position operator $\hat{x}$, which of the following transitions are allowed in first-order perturbation theory? (A) $n \to n + 1$ (B) $n \to n - 1$ (C) $n \to n + 2$ (D) $n \to n - 2$ (E) $n \to n$

複選題18. In a Compton scattering experiment, the spectrum of scattered X-rays displays two distinct peaks: a modified peak at wavelength $\lambda'$ and an unmodified peak at the incident wavelength $\lambda$. The Compton shift is defined as $\Delta \lambda = \lambda' - \lambda$. Which of the following statements regarding this experiment are correct? (A) The modified peak is due to scattering from loosely bound electrons. (B) The Compton shift $\Delta \lambda$ is independent of the wavelength $\lambda$ of the incident X-ray. (C) The Compton shift $\Delta \lambda$ is always smaller than or equal to zero ($\Delta \lambda \leq 0$). (D) The Compton shift $\Delta \lambda$ depends strongly on the atomic number $Z$ of the target material. (E) The Compton shift $\Delta \lambda$ varies with the scattering angle $\theta$.

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