16. For lattice vibration in a crystal with the atomic mass of m, provide the potential energy U(x) in one dimension x to solve the wave function Ψ(x), the oscillation frequency ω, the quantum energy E_n, zero-point energy ε₀, and average energy ⟨E⟩ at temperature T under the condition of the probability function P(E_n) = e^-En/keT to calculate ⟨E⟩ at T ≅ 0 and ⟨E⟩ at very high temperature.
(A) Ψ(x) = (2ⁿn!)^ H_n(x) e-(mωx)^²/2ℏ), ω =, H₀(x) = 1, H₁(x) = 2x
(B)
(C)
(D) U(x) =kx², ⟨E⟩ ≅ ℏω/2 at T ≅ 0, ⟨E⟩ ≅ k_BT at very high T
(E)

1. Consider the two electrons in a helium atom. Which of the following contradicts the Pauli exclusion principle? (A) The two electrons must have opposite spins in the ground state of the atom. (B) The two electrons can have parallel spins in an excited state of the atom. (C) The two electrons can have opposite spins in an excited state of the atom. (D) If the two electrons have parallel spins, they cannot both occupy 2p orbitals. (E) If the two electrons have opposite spins, they can both occupy 2s orbitals.

2. Consider an intrinsic semiconductor. Which of the following is wrong? (A) The conduction band is partially occupied at the temperature T = 300 K. (B) The valence band is totally filled at T = 0 K. (C) Electrons and holes can be introduced into the semiconductor by doping the semiconductor. (D) At T = 0 K, electrons or holes can be introduced into the semiconductor by injecting electrical currents into the semiconductor. (E) At T = 0 K, electrons and holes can be introduced into the semiconductor by exposing the semiconductor to an electromagnetic wave of any frequency.

3. Which of the following statement is correct? (A) The wave function Ψ(x,t) = Asin(kx + ωt) describes a particle moving in the positive x direction. (B) An electron is confined to one-dimensional box of width L and is in its ground state. A proton is confined in another one-dimensional box also of width L also in its ground state. The wave functions have the different wavelength. (C) Consider a particle of mass m in one dimension, moving in a potential V(x) = -k/x3/2, where k ＞ 0. The ground state energy E₀ for the particle in terms of k, m and ℏ is given by. (D) Consider a particle confined to the region 0 ≤ x ≤ L with the wavefunction Ψ(x,t) = Nsin()-iEt and E = 2ℏ²/mL². We need N = to normalize the wavefunction Ψ(x,t). (E) None of the above.

4. In a photoelectric effect experiment on a certain metal, it is observed that incident light of wavelength 413 nm causes electrons to be ejected from the surface of metal with a maximum kinetic energy of 3.2×10⁻¹⁹ J. What is the longest wavelength of light that will eject electrons from this metal? (A) 1.23 nm (B) 12.3 nm (C) 123 nm (D) 1230 nm (E) 1.23 mm

5. Find the drift velocity v_d of the free electron in a copper wire whose cross-sectional area is 1 mm² when the wire carries a current of 1 A. Assume that each copper atom contributes one electron to the electron gas. Each electron has the charge e and in the time t it travels the distance vd t along the wire. The density and molar mass of copper are 8.96 g/cm³ and 63.5 g/mole, respectively. (A) vd = 7.4×10⁻⁵ m/s. (B) vd = 4.7×10⁻² m/s. (C) vd = 1.57×10⁶ m/s. (D) vd = 3.14×10⁻⁶ m/s. (E) vd = 9.4×10⁻² m/s.

6. Consider a gas of atomic hydrogen, which of the followings is not true? (A) The number of possible states that correspond to the quantum number n is 2n². (B) The ground state energy (n = 1) is −13.6 eV. (C) The relative population of the first excited state (n = 2) at 0°C is 4e−405. (D) At the temperature of 10,000°C, we may expect that the number of excited atoms is about 10²¹ for a cubic meter of atomic hydrogen which contains about 2.7×10²⁵ atoms at atmospheric pressure. (E) We may expect to find 1/10 of the atoms in the first excited state at the temperature of 3.3×10⁴ K.

7. Suppose you are in a spaceship travelling through the solar system at a constant speed of v = 1.5×10⁸ m/s relative to Mars. You fire a pulse of laser light out the back of your vessel (i.e. in the direction from where you came from). How fast does an inertial observer on Mars see the pulse leave your ship? (A) 1.5×10⁸ m/s (B) 2.4×10⁸ m/s (C) 3×10⁸ m/s (D) 4.5×10⁸ m/s (E) Something else.

8. Bohr model was derived as a primitive model of Hydrogen atom, which can calculate the orbital radius, wavelength, speed, period, and energies of the electron. Please make required modification on the original Bohr model to calculate r₁, λ₁, v₁, E₁, E₂, and v_α for a muon with m_u = 220 m_e captured by a proton to form a muonic atom. v_α is the photon frequency for transition from E₂ to E₁ (α-line). (A) r₁ = 1.256×10⁻¹³ m, λ₁ = 7.892×10⁻¹³ m, v₁ = 2.35×10⁶ m/s, E₁ = −3668 eV, E₂ = −917 eV, vα = 6.651×10¹⁷ Hz (B) r₁ = 1.865×10⁻¹³ m, λ₁ = 1.172×10⁻¹² m, v₁ = 2.35×10⁶ m/s, E₁ = −3164 eV, E₂ = −791 eV, vα = 5.737×10¹⁷ Hz (C) r₁ = 2.185×10⁻¹³ m, λ₁ = 1.373×10⁻¹² m, v₁ = 2.22×10⁶ m/s, E₁ = −2816 eV, E₂ = −704 eV, vα = 5.106×10¹⁷ Hz (D) r₁ = 2.694×10⁻¹³ m, λ₁ = 1.693×10⁻¹² m, v₁ = 2.187×10⁶ m/s, E₁ = −2672 eV, E₂ = −668 eV, vα = 4.845×10¹⁷ Hz (E) r₁ = 3.068×10⁻¹³ m, λ₁ = 1.928×10⁻¹² m, v₁ = 2.198×10⁶ m/s, E₁ = −2316 eV, E₂ = −579 eV, vα = 4.2×10¹⁷ Hz

9. For an electron with a de-Broglie wavelength of 2.425×10⁻¹² m, please find the correct the momentum p, kinetic energy KE, rest energy E₀, total energy E, and velocity v of the electron and phase velocity vp. (A) p = 2.7324×10⁻²² kg·m/s, v = 2.9995×10⁸ m/s, KE = 4.0978×10⁻¹⁴ Joule (B) E₀ = 8.1876×10⁻¹⁴ Joule, KE = 4.0978×10⁻¹⁴ Joule, E = 1.2285×10⁻¹³ Joule (C) p = 511 KeV/c, KE = 212 KeV, v = 2.12×10⁸ m/s, vp = 4.2388×10⁸ m/s ＞ c (D) p = 511 KeV/c, KE = 212 KeV, E = 723 KeV, v = 2.73×10⁸ m/s, vp = 3.2925×10⁸ m/s ＞ c (E) p = 511 KeV/c, KE = 255.8 KeV, v = 2.9995×10⁸ m/s, vp = 2.9965×10⁸ m/s

10. You do the photoelectric effect with a big chunk of Gold (work function= 5.1 eV), which you happened to have to lying around in your room. You find that with the light source you have at hand, you get a stopping potential of 1.1 V. What is the wavelength of your light source? (A) 150 nm (B) 200 nm (C) 300 nm (D) None of the above. (E) Cannot tell, because there is not enough information to calculate this.

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