所屬科目：研究所、轉學考(插大)-近代物理學
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)
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) $10.6 \mu \mathrm{m}$
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.
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.
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.
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.
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.
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 $3 \times 10^{4} \mathrm{~m/s}$.
17. A beam of neutrons is traveling at a speed of $v = 800 \, \mathrm{m/s}$ and is directed through a single slit of width $a = 1.0 \, \mu \mathrm{m}$. Given the Planck's constant $h = 6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}$ and the neutrons have mass $m_{n} = 1.675 \times 10^{-27} \, \mathrm{kg}$. Consider the neutrons as free quantum particles and their wave-like behavior. Which of the following statements are correct? (A) The de Broglie wavelength of the neutrons is approximately $4.945 \times 10^{-10} \, \mathrm{m}$. (B) The first minimum of the diffraction pattern on a screen far away from the slit occurs at an angle $\theta$ given by $\sin \theta = \lambda / a$. (C) The phase velocity of the neutron wave is greater than the speed of particles. (D) The group velocity of the neutron wave packet is equal to the speed of the particles. (E) The uncertainty in the neutron's transverse momentum due to the slit is approximately $6.63 \times 10^{-28} \, \mathrm{kg} \cdot \mathrm{m/s}$.
18. Which of the following statements are correct? (A) According to the Bragg's law, when monochromatic X-rays of wavelength $\lambda = 0.154\,\mathrm{nm}$ are incident on the crystal with lattice spacing $d = 0.5\,\mathrm{nm}$, the first-order diffraction angle $\theta$ is approximately $18^{\circ}$. (B) A spectrometer on Earth with a detection limit of $0.5\,\mathrm{nm}$ can be used to detect the difference between classical and relativistic Doppler's shifts of a star moving toward Earth at a speed of $6\times 10^{6}\,\mathrm{m/s}$ and emitting light at a wavelength of $500\,\mathrm{nm}$. (C) For a free particle in one dimension with the time-dependent Schrödinger wave function $\psi(x,t) = Ae^{i(kx - \omega t)}$, the probability density is independent of time. (D) In the Compton scattering, a photon scattered by a free electron will generally emerge with a longer wavelength if it is deflected through a larger angle. (E) A particle whose wave function is a perfect plane wave has a precisely defined momentum.
19. Consider 2 quantum, spin-1/2, non-interacting electrons in a 1D box. Let the box width L. Let n = quantum number of the one-electron energy eigenstate with, for example, n = 0 for the ground state, n = 1 for the 1st excited state, and so on. Denote the two-electron state by $(n_1, n_2)$, where $n_1$ and $n_2$ are the quantum numbers of the one-electron states occupied. Which of the following statements are correct, regarding the two-electron state? (A) The ground state is given by $(n_1 = 0, n_2 = 0)$. (B) The first excited state is given by $(n_1 = 0, n_2 = 1)$. (C) The two spins in the state $(n_1 = 0, n_2 = 0)$ are anti-parallel. (D) The two spins in the state $(n_1 = 0, n_2 = 1)$ can be either parallel or anti-parallel. (E) The two spins in the state $(n_1 = 1, n_2 = 1)$ can be either parallel or anti-parallel.
20. Which of the following statements are wrong, for a particle totally confined in a 1D asymmetric, piecewise constant potential V(x). (A) The ground state wave function is even in parity. (B) The probability vanishes at infinity. (C) The first excited state is odd in parity. (D) The quantization leads to discrete energy levels. (E) The ground state is non-degenerate.
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