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) $E_{F}\propto n^{1 / d}$

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

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 $(\hat{S}_z)$. 

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