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)

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.

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.

複選題12. Which of the following statements are correct? (A) In the classical limit, high temperature or low density, both Bose–Einstein and Fermi–Dirac statistics approach the Maxwell-Boltzmann distribution. (B) Helium-4 atoms obey Fermi-Dirac statistics. (C) Photons obey Bose-Einstein statistics and have a chemical potential equal to zero. (D) Bose-Einstein statistics are commonly used to describe phonons in semiconductors because phonons are integer-spin quasiparticles and their number is not conserved. (E) In heavily doped semiconductors, if the carrier concentration approaches or exceeds the effective density of states, electrons or holes can no longer be described by Maxwell–Boltzmann statistics and must instead be treated using Fermi-Dirac statistics.

複選題13. Which three of the following statements correctly describe the hydrogen atom, the harmonic oscillator, and the particle in a box in quantum mechanics? (A) The energy quantization of a harmonic oscillator obtained from the Schrödinger equation reduces exactly to Planck's quantization formula; however, Max Planck did not provide a physical interpretation for this quantization. (B) For a particle in a two-dimensional box, the numbers of ground and first excited states are identical as long as the box is square. (C) For a harmonic oscillator, the ground-state energy satisfies the lower bound imposed by the Heisenberg uncertainty principle because its wavefunction is Gaussian. (D) For the hydrogen atom, the principal quantum number specifies the total energy and the number of quantum states, while the spatial symmetry of the wavefunction must be further determined by the orbital and magnetic quantum numbers. (E) In all three types of potentials, the wavefunction must vanish at infinity if the particles form a bound state under the influence of these potentials.

複選題14. Which three of the following statements are correct? (A) The Einstein model for calculating the specific heat of solids does not account for the polarization of lattice vibrations, and its predictions are consistent with experimental results at low temperatures. (B) In both the photoelectric effect and blackbody radiation, the number of photons is quantized and is not conserved. (C) The cosmic microwave background radiation can be approximated as blackbody radiation in order to estimate the temperature of the universe. (D) For quantum tunneling, the tunneling probability is always 100% as long as the particle's kinetic energy is greater than the potential barrier, which reduces to the classical result. (E) In the hydrogen atom, the orbital quantum number is related not only to the rotational kinetic energy but also to the number of radial nodes in the wavefunction.

複選題15. Now, we have two electrons whose orbital angular momentum quantum numbers are 1 and 3, respectively. Choose the correct answers. (A) The possible TOTAL angular momentum quantum number J can be 1 (B) The possible ORBITAL angular momentum quantum number L can be 1 (C) The possible SPIN angular momentum quantum number S can be 1  (D) The possible MAGNETIC quantum number of total angular momentum number $m_{j}$ can be $\frac{1}{2}$ (E) The possible spin multiplicities for this two-electron system are singlet and triplet.

複選題16. We usually need to use the concept of "effective mass" to analyze and predict the properties of electronic and optoelectronic devices. It can be calculated by $\frac{1}{m^{*}} = \frac{1}{h^{2}}\frac{\partial^{2}E(k)}{\partial k^{2}}$, where $m^{*}$ is the effective mass, and $E(k)$ is the energy of the carrier as a function of the wave vector $k$. The effective mass of carriers in a material can be measured using the cyclotron resonance method. In short, this method is performed by applying a magnetic field to the material and irradiating it with electromagnetic waves, assuming that the absorption frequency of the sample matches the cyclotron frequency of the carriers in the magnetic field. Hint: You may start the derivation from the classical equation of motion of a charged particle in a magnetic field. The magnetic force is $qvB$, where $q$ is the carrier charge, $v$ is its velocity, and $B$ is the magnitude of the magnetic field, and it provides the centripetal force for circular motion ($F = \frac{m^{*}v^{2}}{r}$). Please choose the correct answer. (A) The idea of effective mass is only needed for semiconductors. For metals, the free electron model is sufficient. (B) Effective mass can be less than $m_e$, where $m_e$ is the mass of an electron. (C) The effective mass is determined by the curvature of the energy band E(k). (D) When the magnetic field is $0.15\mathrm{T}$ and the absorption frequency $= 2.1\times 10^{10}\mathrm{Hz}$, the effective mass of an electron is $0.3m_e$. (E) If the absorption frequency is $1.4 \times 10^{10} \mathrm{~Hz}$ and the radius is $3.4 \times 10^{-7} \mathrm{~m}$, the velocity of the electron should be $3 \times 10^{4} \mathrm{~m/s}$.

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