6. Consider a particle moving in a 2D square quantum' well, with well width = L and well depth = ∞. Take potential well bottom = 0. Which of the following statements is correct, in regard to state?
(A) The ground state energy is four times as large as that in a 1D quantum well of identical width and depth.
(B) The ground state energy is 1/4 of the first excited state energy.
(C) The ground state wave function is given by the sum F₀(x) + F₀(y), ((x,y) = particle position), where F₀ is the ground state wave function in a 1D quantum well of identical width and depth.
(D) The ground state wave function vanishes at the well center.
(E) The ground state wave function is even with respect to the inversion operation (x,y) → (-x,-y).

2. Which of the following statements is wrong, in regard to the physics of a 1D oscillator in the quadratic potential energy ? Let n denote the state quantum number of oscillation, with n = 0 for the ground state and n = 1, 2, for excited states. (A) The classical ground state energy is given by E0 = 0. (B) Quantum energy levels {En's} are equally spaced. (C) Oscillation frequencies are identical for classical and quantum oscillators. (D) For large n, the quantum description approaches the classical one. (E) The ground state wave function (Note: xtis the classical turning point where the kinetic energy vanishes.)

3. Assuming that the conduction electrons in a cube of a metal on edge 1 cm behave as a free quantized gas, what is the number of states which are available in the energy interval 4.00-4.01 eV, per unit volume? (A)(B) (C) (D) (E)

4. What is the depletion-layer width for a p-n junction with zero bias in germanium, given that the impurity concentrations are, respectively, at T = 300 K, relative permittivity εr = 16 and contact potential difference Vo = 0.8 V? (A) 0.0725 μm (B) 0.145  μm (C) 0.29 μm (D) 0.58  μm (E) 1.16  μm

複選題5. For a muon with mμ = 200 me captured by a proton to form a muonic atom, please calculate the orbital radius, wavelength, speed, and energies at ground state n= 1: r1, λ1, v1, E1, and photon wavelength  λa for transition from E2 to E1 (c-line) (A) (B) (C) v1= 2.1872x106m/s (D) E1=-2505 eV (E)

7. Electrons with energy of 2 eV are separately incident on 10 eV barriers with 0.5 nm and I nm wide. What are their respective transmission probabilities? (A) (B) (C)(D) (E)

8. In terms of a particle trapped in a box with the length of L, which of following statements is wrong? (A) The normalized wave functions of the particle are (B) At ground state, the probability of the particle which is found between 0.45 L and 0.55 L is about 10%. (C) At first excited state, the probability of the particle which is found between 0.45 L and 0.55 L is about 0.65%. (D) At first excited state, the probability of the particle which is found at x = 0.5 L is zero. (E) At a particular place in the box, the probability of the particle may be different for different quantum numbers.

複選題9. Which of the following statements are correct concerning the quantum energy eigenstate of a free particle? Let E = energy, p = momentum, and k = wave vector of the state.【複選題】(A) E's of all eigenstates form a continuous spectrum. (B) gives the speed of a moving wave packet formed by mixing the eigenstates with (E, p) around (Ec, Pc). (C) A particle with k will behave approximately classically, when hitting a wall with an opening of the size L＞＞ 1/|k|. (D) A mixed state formed of two eigenstates such as exp, where ki and k2 differ but|k1|=| k2|, is not an eigenstate of momentum but is that of energy. (E) If one measures the momentum of mixed state exp(ik●r) + exp(-ik●r) for many times by performing the measurement on a very large number of identical copies of the state, the distribution of measured values will have a width of the order of |k|.

複選題10. Which of the following experimental observations that challenged classical physics around 1900?【複選題】 (A) Blackbody radiation. (B) Electromagnetism and the origin of magnetic fields, which did transform from one coordinate system to another. (C) Line spectra of hydrogen and other atoms. (D) The photoelectric effect. (E) Compton scattering.

複選題11. Consider the quantum system of two identical, spin-1/2 particles moving in a 1D infinite quantum well, in the presence of a uniform magnetic field B. Let x1 and x2 denote the two particle positions, szl and s22 the two particle spins in the B direction, Ez = Zeeman energy, and dE = energy spacing between the ground and first excited states of one particle in the well at B = 0. Which of the following statements are correct?【複選題】 (A) When B is turned off, the ground state of the two-particle system consists of two particles of opposite spins. (B) When B is turned on, the ground state of the two-particle system consists of two particles of parallel spins, if Ez ＞＞ dE. (C) At B = 0, the wave function of the two-particle system is of the determinant form (D) The Pauli exclusion principle implies that the wave function of the two-particle system must al ways vanish if (E) All of the above are true.

複選題12. A muon has a half lifetime and a mass of 200 me. This muon is moving with a speed of u = 0.6 c. Please calculate mean lifetime τ0, the traveling distance d measured at rest, during mean lifetime To the relativistic momentum P, the de-Broglie wavelength λd, the total energy E and kinetic energy (KE) in the unit of eV, the wavelengths of a photon with the same kinetic energy or the same momentum P, λp【複選題】 (A) The mean lifetime and traveling distance measured at rest is (B) The relativistic momentum is P = 76.66 MeV/c (C) The de-Broglie wavelength is (D) The total energy and kinetic energies are E = 165.77 MeV, KE = 35.55 MeV (E) The wavelengths of photon with the same P or KE: