複選題

Problem 17 Which of the statements are CORRECT about a particle confined in the 1D infinite potential well V(x) with V(x) = 0 in the well (with well width = W).
(A) The ground state energy aligns with the bottom of the well.
(B) For the first excited state, the uncertainty in the position is of the order of W.
(C) Let 1 = wavelength of the ground state standing wave. Then I is of the order of W.
(D) Let p = momentum of the ground state. The p is of the order of 'h/W, with the uncertainty in p being of the same order as p (h = Planck constant).
(E) None of the above.

Problem 1What physical situation (in one dimension) does the following time-dependent Schrödinger equationdescribe?where k and e are real, non-zero constants.(A)A particle traveling up a ramp along the x-axis.(B) A particle in a finite square well potential.(C) A particle in an infinite square well potential.(D) An electron moving along the x-axis in the electric field of a proton at x=0.(E) None of the above.

Problem 2Find the correct statement:(A) The Fermi-Dirac distribution can be used to in the analysis of the properties of photons.(B) The peak wavelength of blackbody radiation is inversely proportional to temperature (1/T).(C) The spin of a photon is 1/2.(D) In a quantized system, each quantum state can only contain one photon.(E) At room temperature, the average kinetic energy of gas molecules is proportional to T2.

Problem 3Consider electron states of a particle in a finite quantum well, with the potential energy V(x) definedby V(0 ＜x ＜ W) = 0 in the well and V(x) = V0 ＞ 0 outside the well. Which of the statements is CORRECT?(A) An eigenstate in this case always describes a bound state in space.(B) The first excited state has an energy E1 with E1 ＜ Vo always.(C) The wave function of ground state can sometimes oscillate with nodal points where the wave function vanishes.(D) For any bound eigenstate in the system, the probability density is nonvanishing in the well and vanishing outside the well.(E) None of the above.

Problem 4Which quantum number (n, l, mi, ms) is not a possible set of a 2p electron?(A) (2, 1, +1, +1/2)(B) (2, 1, +1, -1/2)(C) (2, 1, 0, +1/2)(D) (2, 0, -1, +1/2)(E) (2, 1, -1, -1/2)

Problem 5According to band theory, beryllium, an alkaline earth metal, is predicted to behave as an insulator.However, it acts as a conductor due to:(A) The presence of a half-filled 2s band.(B) The overlap between the filled 2s band and the empty 2p band.(C) The presence of an empty 2p band.(D) The overlap between the filled 2p band and the empty 3s band.(E) The presence of a half-filled 3s band.

Problem 6 Consider the transmission of an incident electron with energy E through the one dimensional step potential barrier V(x) with V(x ＜ 0) = 0 outside the barrier, and V(x ＞ 0) = Vo in the barrier (V0 ＞ 0). Which of the following statements is CORRECT about the transmission coefficient T? (A) If the incident electron comes from x = +infinity, with E＞ V0,then the corresponding T = 1. (B) If the incident electron comes from x = - infinity, with E ＞V0, then the corresponding T = 1. (C) If the incident electron comes from x = - infinity, with E＜V0, then it has a nonvanishing probability in the barrier as well as T ＞ 0 due to quantum tunneling. (D) Replace V(x) by a potential barrier of finite width, with V(0 ＜ x ＜W) = Vo in the barrier and V(x) = 0 outside the barrier. An incident electron with E＜V0 has T ＞ 0 due to quantum tunneling. (E) None of the above.

Problem 7An electron is bound in a potential well. The wave function of the electron is:Y(x,t) = v(x)e-iEtn, where φ(x) is shown as below and E is a real number. Which one ofthe following potential wells would fit best to this wave function?(A)(B)(C)(D)(E)

Problem 8 A one-dimensional infinite potential well locating between -0.5 nm to 0.5 nm and contains 10 electrons. It has 10 electrons inside. What is the least energy, in eV, the incident light must have in order to excite a ground-state electron (i.e., at the lowest n quantum value) in this system to the lowest higher state it can occupy when T=0? (A) ~8.7 eV, (B) ~13.6 eV, (C) ~5.61 eV, (D) ~13.2 eV, (E) ~44.9 eV.

Problem 9In Bohr's hydrogen model, if the Coulomb interaction between the electron and the nucleus changes to a power-law potential: V(r) = krN, where k and N are positive constants. Applying Bohr's hypothesis to calculate the energy of hydrogen atom, the energy will be proportional to the quantumnumber n as(A)(B) (C)(D) , and (E) n-2.

Problem 10Which one of the following statements about free bosons, fermions, and classical particles is correct?(A) Bosons possess integer spin and follow the Maxwell-Boltzmann distribution in the low-temperature regime.(B) Fermions are indistinguishable particles that obey the Pauli exclusion principle and follow the FermiDirac distribution in all temperature range.(C) Classical particles can either be distinguishable or indistinguishable depending on the type of atoms or particles.(D) In the low-temperature limit, bosons and fermions cannot occupy the same quantum state, in contrast to classical particles.(E) In Bose-Einstein condensation, particles follow the Bose-Einstein distribution, with most particles occupying either the same excited state or the ground state. This phenomenon can be observed macroscopically.

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