7. Which of the following statements about free particles is correct?
(A) Based on the Einstein-de Broglie relation, the particle speed can be determined by the ratio of the de Broglie angular frequency to the wavenumber.
(B) Wave-particle duality becomes apparent when the particle dimension is comparable to its matter-wave wavelength.
(C) The Schrödinger equation can be used to describe particles at small length scales while accounting for the effects of special relativity.
(D) For free particles, both the uncertainties in momentum and position are finite, and their product has a lower bound that satisfies Heisenberg's uncertainty relation.
(E) In the Schrödinger equation, the product of the group and phase velocities does not exceed the speed of light, since the speed of information propagation cannot be greater than the speed of light.

1. Consider a particle in a 2D rectangular box. Let the box widths be L and 2L, respectively, in the x- and y-dimensions. Which of the following expressions is the correct first excited state energy? $(h = \text{Planck constant}, m = \text{particle mass})$ (A) $\frac{h^2}{8mL^2}$ (B) $\frac{h^2}{16mL^2}$ (C) $\frac{h^2}{4mL^2}$ (D) $\frac{h^2}{2mL^2}$ (E) $\frac{h^2}{32mL^2}$

2. Consider the EM wave mode with wave vector k and frequency f, in a cavity of temperature T. The average number of photons in the mode, in the limit where $k_{B}T \gg hf$, is given by $(k_{B} = \text{Boltzmann constant}, h = \text{Planck constant})$ (A) $\left[\frac{k_B T}{h f}\right]^2$ (B) $\frac{1}{\left[\exp\left(\frac{k_B T}{h f}\right) - 1\right]}$ (C) $\frac{k_B T}{h f}$ (D) $\exp\left(\frac{-hf}{k_B T}\right)$ (E) $\frac{1}{\left[\exp\left(\frac{hf}{k_B T}\right) + 1\right]}$

3. A muon has a mean lifetime of $2.20 \times 10^{-6}$ s at rest. If it travels at a speed of 0.99c, how far can it travel before decaying? Given that the light velocity $c = 3 \times 10^{8} \, \mathrm{m/s}$. (A) $65\mathrm{m}$ (B) $653\mathrm{m}$ (C) $4632\mathrm{m}$ (D) $6534\mathrm{m}$ (E) $92\mathrm{m}$

4. A beam of ultraviolet light with wavelength $250\mathrm{nm}$ shines on a metal surface and ejects photoelectrons with a maximum speed of 0.004c. Which of the following statements is correct? Given that the light velocity $c = 3\times 10^{8}\mathrm{m / s}$, the Planck's constant $h = 6.626\times 10^{-34}\mathrm{J\cdot s}$, the elementary positive charge $q = 1.6\times 10^{-19}\mathrm{C}$, and the electron mass $m_{e} = 9.11\times 10^{-31}\mathrm{kg}$. (A) The work function of the metal is $0.87\mathrm{eV}$. (B) A wavelength larger than $1.45\mu \mathrm{m}$ can eject electrons from this metal. (C) The photon energy is $4.10\mathrm{eV}$. (D) The threshold frequency is about $1.2 \times 10^{15} \mathrm{~Hz}$. (E) The kinetic energy of the photoelectron is $6.56 \times 10^{-19} \mathrm{eV}$.

5. Find the correct description: (A) The photon energy of light from the Paschen series (transitions from n ≥ 4 to n = 3) of the hydrogen atom is sufficient to induce the photoelectric effect in silver (work function $\approx 4.7$ eV). (B) For n = 4, the radius of the hydrogen atom in the Bohr model is $4a_{0}$, where $a_{0}$ is the Bohr radius. (C) The Balmer series, corresponding to electronic transitions from any higher level ($n \geq 3$) to n = 2, mainly lies in the visible region of hydrogen's atomic spectrum. (D) Rutherford's scattering experiment suggested that the positive and negative charges are uniformly distributed throughout the atom. (E) In the Bohr model, electrons continuously lose energy while orbiting the nucleus and eventually spiral into it.

6. Johnny just found a box containing multiple wafers (thickness: 380 µm). On the label attached to the box, he found that the wafers are made of different materials: InP, GaP, GaAs, Ge, and Si, respectively. His advisor asked him to sort them. He randomly picked up one of them and discovered that it allows visible light to pass through under ambient light. Which wafer could it be? (A) InP (bandgap ≈ 1.27 eV) (B) GaP (bandgap ≈ 2.26 eV) (C) GaAs (bandgap ≈ 1.42 eV) (D) Ge (bandgap ≈ 0.66 eV) (E) Si (bandgap ≈ 1.12 eV)

8. Which of the following statements correctly describes quantum particles under time-independent, externally applied potentials? (A) For a particle in a box, the wavefunction forms standing waves whose energy is proportional to the sum of the squares of the numbers of antinodes. (B) For particles in one-dimensional space, the wavefunction is always real rather than complex as long as the potential is finite. (C) For a harmonic oscillator, the energy difference between successive quantum states depends on the number of nodes in the wavefunction. (D) In the tunneling effect, particles experience a Dirac delta potential, and the tunneling probability is zero because the potential becomes infinite at a specific point. (E) If the potential is finite but discontinuous at some points, the derivative of the wavefunction may or may not be smooth, depending on the functional form of the potential.

9. Light with a wavelength of $434\mathrm{nm}$ is observed from a hydrogen discharge lamp. Which initial energy level (higher level) most likely produced this emission, assuming the electron transitions to a lower level? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

10. Assuming the human body temperature is approximately $37^{\circ}\mathrm{C}$, which of the following wavelengths is closest to the peak wavelength of its thermal radiation? (A) $9.0 \mu \mathrm{m}$ (B) $9.4 \mu \mathrm{m}$ (C) $9.8 \mu \mathrm{m}$ (D) $10.2 \mu \mathrm{m}$ (E) $10.6 \mu \mathrm{m}$ 

複選題11. Which of the following statements are correct? (A) In both blackbody radiation and the photoelectric effect, the assumption of energy quantization plays a crucial role; however, in Rutherford scattering, quantization is not a necessary assumption. (B) The photoelectric effect and blackbody radiation both rely on Planck's quantization hypothesis, but only the photoelectric effect requires the use of photon momentum to explain the observed phenomena. (C) In the photoelectric effect, increasing the intensity of the incident light increases the maximum kinetic energy of the emitted electrons. (D) In Rutherford scattering, the interaction between an $\alpha$-particle and the atomic nucleus is primarily governed by a short-range force. (E) As the energy of the incident $\alpha$-particles increases, the probability distribution of scattering angles shifts toward smaller deflection angles.

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