114年 - [無官方正解]114 國立中山大學碩士暨碩士專班招生考試_物理系碩士班：近代物理#138402

1. What is the energy carried by a photon with a wavelength of 650 nm? (A) 5.1 × 10⁻²⁰ eV (B) 3.1 × 10⁻¹ eV (C) 3.2 × 10⁻¹⁹ eV (D) 2.0 eV (E) 6.6 × 10⁻⁹ eV

2. Which of the following quantities is invariant under Lorentz transformation? (A) Length (B) Time (C) Energy (D) Momentum (E) Mass

3. Two rockets approach Earth from opposite directions at a high speed v. What is the speed of a rocket as measured by the observer on the other rocket? (A) (B) 2v (C)(D) (E)

4. What is the expectation value of momentum squared ⟨p²⟩ for a particle described by the normalized wavefunction Ψ(x,t) = , where E is a real constant and a ＞ 0, in the position space? (Hint: dxe^-αx²= for α ＞ 0) (A) (B) iℏ (C) (D) (E)

5. What is the energy carried by a photon with momentum ? (A)(B)(C) (D)(E)

6. Which of the following distributions describes the statistical behavior of an ensemble of spin-1/2 identical particles at a low-temperature T? (k_B is Boltzmann's constant and E_F is Fermi energy.) (A)(B)(C) (D) (E)

7. Which of the following statements about superconductivity is correct? (A) All materials become superconducting below 30 K (B) The resistance of a superconductor is zero (C) Superconductivity violates the first law of thermodynamics (D) Superconductivity violates the second law of thermodynamics (E) All the above are incorrect

8. Which of the following pairs of Nobel laureates and their major reasons for being awarded is incorrect? (A) Niels Bohr --- atomic model (B) Louis de Broglie --- wave-particle duality (C) Albert Einstein --- quantum entanglement (D) John Bardeen --- superconductivity (E) Richard Feynman --- quantum electrodynamics

9. Which of the following discoveries did not contribute to the development of quantum mechanics? (A) Hydrogen spectrum (B) General relativity (C) Energy quanta (D) Wave-particle duality (E) All the above did

10. Which of the following is not a property of a laser? (A) High monochromaticity (B) High coherence (C) High directivity (D) High propagation speed (E) High intensity

11. If a quantum particle sees a potential shown in Figure 1 (a double-well potential), which of the following probability densities is most likely to be the ground state probability density?

(A)(B)(C)(D) (E)

12. Given a particle with energy E in a finite potential U(x), as shown in Figure 2 (with regions I through V, where Region III has U ＞ E), what is the form of the particle's wavefunction in Region III?

(A) ce^κx, where κ ∈ ℝ (B) cx² (C) c₊e^κx+ c₋e^-κx, where κ ∈ ℝ (D) ce^-κx, where κ ∈ ℝ (E) c₊sin(kx) + c₋cos(kx), where k ∈ ℝ

13. A photon is directed toward a stationary particle of mass m with momentum = pê_x. After the collision, the photon reflects along the same path, i.e., = -p'ê_x. What is p'? (A) p (B)(C)(D) p - mc (E) 0

14. Which of the following masses is the smallest? (Hint: A neutron can decay into a proton, an electron, and an antineutrino.) (A) The mass of a Helium-4 atom (a stable atom consisting of two protons, two neutrons, and two electrons) (B) The total mass of an alpha particle (a helium ion consisting of two protons and two neutrons) and two electrons (C) The total mass of two protons, two neutrons, and two electrons (D) The total mass of one proton, three neutrons, and one electron (E) All the above are the same

15. What is the magnitude of the spin angular momentum of a spin-1/2 particle? (A) s = (B) s = (C) s = (D) s = (E) s = ℏ

16. Alice and Bob are making measurements on a two-qubit system described by the quantum state |ψ⟩ = 0.6|1⟩_A ⊗ |0⟩_B + 0.8|0⟩_A ⊗ |1⟩_B, where |m⟩_A and |n⟩_B are Alice's and Bob's measurement eigenstates, with eigenvalues m and n, respectively. If Alice makes the measurement first and the outcome is 1, what possible outcomes can Bob obtain, and what are the probabilities? (A) Both 0 and 1 with probability 36% and 64%, respectively (B) Both 0 and 1 with probability 64% and 36%, respectively (C) Both 0 and 1 with probability 50% and 50%, respectively (D) Only 1 (i.e., with probability 100%) (E) Only 0 (i.e., with probability 100%)

17. Which of the following forces primarily underlies the mechanism of nuclear power? (A) Gravitational force (B) Electric force (C) Magnetic force (D) Weak force (E) Strong force

18. Let the wave function of two electrons at t = 0 be Ψ(x₁,s₁;x₂,s₂), where x₁ and x₂ are the positions, and s₁ and s₂ are the spins of the electrons. Given that the wave function can be factorized into spatial and spin parts, namely, Ψ(x₁,s₁;x₂,s₂) = ψ_a(x₁,x₂)χ_a(s₁,s₂) + ψ_b(x₁,x₂)χ_b(s₁,s₂), where χ_a(s₂,s₁) = -χ_a(s₁,s₂) and χ_b(s₂,s₁) = χ_b(s₁,s₂), which of the following statements is incorrect? (A) ψ_a(x₁,x₂) = -ψ_a(x₂,x₁) (B) ψ_b(x,x) = 0 (C) Ψ(x₂,s₂;x₁,s₁) = -Ψ(x₁,s₁;x₂,s₂) (D) ψ_a(x₂,x₁)ψ_b(x₂,x₁) = -ψ_a(x₁,x₂)ψ_b(x₁,x₂) (E) All the above are correct

19. If a muon with a lifetime of 2.0 μs is traveling at a speed of 0.60c relative to Earth, how far does it travel during its lifetime as observed by a stationary observer on Earth? (Gravitational effects can be neglected.) (A) 2.9 × 10² m (B) 3.6 × 10² m (C) 4.5 × 10² m (D) 6.0 × 10² m (E) 9.0 × 10³ m

20. Given a three-dimensional wave function Ψ(x,y,z,t) = A exp, where E and a are positive constants with units of energy and length, respectively, which of the following values of A ensures the wave function is normalized? (A) (B) (C)(D) (E)