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複選題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.

複選題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.

複選題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.

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