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

1. Let $S$ and $S'$ be two inertial frames of reference. $S'$ moves relative to $S$ with velocity $v$ in the $z$-direction. In $S$, two events occur at the same location with a time separation of 4s. What is the spatial distance of these two events in $S'$, given that they occur with a time separation of 5s in that frame? (A) 1cs (B) 2cs(C)2.42cs (D) 3cs (E) 4cs

2. Two particles, each of mass $m$, move directly toward one another at a speed of $v = 0.866c$. After the head-on collision, all that remains is a new particle of mass $M$. What is the mass of this new particle? (At $v = 0.866c$, $\sqrt{1 - v^2/c^2} = 0.5$). (A) $M = 2.0m$ (B) $M = 4.0m$ (C) $M = m$ (D) $M = 1.5m$ (E) $M = 5.0m$

3. The momentum of a photon is $2.000 \, \mathrm{MeV} / c$. What is its total energy? (A) 2.000 MeV (B) 2.200 MeV(C) 4.000 MeV (D) 0.970 MeV (E) 3.100 MeV

4. Certain surface waves in a fluid travel with phase velocity $v_{p} = \sqrt{b / \lambda}$, where $b$ is a constant and $\lambda$ is the wavelength. Find the group velocity of a packet of surface waves, in terms of the phase velocity $v_{p}$. (A) $v_{p}$ (B) $3v_{p} / 2$ (C) $3v_{p} / 4$ (D) $2v_{p}$ (E) $5v_{p} / 2$

5. Including the electron spin, what is the degeneracy of the $n = 5$ energy level of hydrogen? (A) 25 (B) 16 (C) 24 (D) 50 (E) 60

6. In a certain region of space, a particle of mass $m$ is described by the wave function $\psi(x) = Cxe^{-bx}$ where $C$ and $b$ are real constants. The potential $V(x)$ in this region is proportional to $1/x$, with no constant term $(V(x) \propto 1/x)$. By substituting this into the time-independent Schrödinger equation, find the energy of the particle, assuming the energy is constant throughout the region and independent of $x$. (A) $\hbar^2 b^2 / m$ (B) $-\hbar^2 b^2 / m$ (C) $\hbar^2 b / m$ (D) $-\hbar^2 b^2 / 2m$ (E) $\hbar b / 2m$ 

7. Which of the following physical quantities has the same dimension as the Planck constant? (A) momentum (B) angular momentum (C) energy (D) power (E) photon frequency

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) $\langle H\rangle = E_1 + \sqrt{2} E_2$ ; $|\Psi (x,t)|^2 = \psi_1^2 +2\psi_2^2 +2\sqrt{2}\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$

115年 - 115 台灣聯合大學系統_碩士班招生考試_物理類：近代物理#139678

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