課程名稱︰近代物理 課程性質︰電機系複選必修 課程教師︰林清富 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰2018/01/11 考試時限（分鐘）：180 試題 : 1/2 Final (two pages) The following formulas or values may be needed for the problems. 2 1 ∂ ∂f 1 ∂ ∂f 1 ∂^2 f a.▽ f= ── ──(r^2──)+───── ──(sinθ──)+─────── ─── r^2 ∂r ∂r r^2 sinθ ∂θ ∂θ r^2 sin^2(θ) ∂φ^2 b. wave functions for Hydrogen atom: 1 -r/a_0 ─────── e for n = 1 √π a_0^(3/2) 1 -r/2a_0 r ───────── e (2 - ──) for n = 2, l = 0 4√(2π) a_0^(3/2) a_0 1 -r/2a_0 r ───────── e (──) cosθ for n = 2, l = 1, m = 0 4√(2π) a_0^(3/2) a_0 1 -r/2a_0 r ±jφ ──────── e (──) sinθe for n = 2, l = 1, m = ±1 8√π a_0^(3/2) a_0 _2 2 -11 c. Bohr radius a_0 = 4πε_0h / me = 5.292× 10 m. 4 2 3 7 -1 d. Rydberg constant R = me / 8ε_0 ch = 1.097×10 m . _ eh -5 e. Bohr magneton μ = ── = 5.788×10 eV / Tesla; B 2m -23 Boltzmann's constant k = 1.381x10 J/K; -34 Planck's constant h = 6.63x10 J‧s; -31 -27 electron mass m = 9.1x10 kg; photon mass m = 1.67x10 kg. 1. Consider a particle in a spherical potential well of finite depth: V(r) = 0 0＜r＜R V(r) = K r≧R where K＞0. (a) Is the orbital angular momentum quantized in this system? If yes, how is it quantized? (i.e. L^2 or L=?) (b) If three electrons are in such a spherical potential well, please assign the possible quantum numbers to each electron as the system is in the ground state. (Ignore the Coulomb interaction between electrons, but consider that electrons are Fermions.) (10) 2. A photon with energy in the visible region (between about 400 and 700 nm) causes the transition n → n+1 in doubly ionized lithium, ++ Li -- a hydrogen-like system with a single electron and a central charge equal to 3e. What is the lowest value of n for which this could occur? (10) 3. When l = 2. s = 1/2, please calculate the possible angles between the directions of L and S. (10) 4. Please explain the difference between the fluorescence and the phosphorescence. (10) 5. D is deuterium, the isotope of H. D has an atomic mass approximately twice that of ordinary hydrogen. Calculate the energies of the four lowest non-zero rotational 2/2 energy states of H_2 and D_2. (10) 6. Please prove that 2s and 2p wave functions for hydrogen Schrodinger's equation are orthogonal. (10) 7. The bandgap energy of GaAs is 1.42 eV. Assuming that the Fermi energy is at the middle of the bandgap, (a)calculate the probability for an energy state in the conduction band to be occupied by an electron at room temperature. (10) (b)Does the GaAs semiconductor have the same number of electrons and holes at room temperature? Why? (10) 8. Please compare the transition energies between vibrational modes and rotational modes for the molecule of CO. Which one has a larger value? Why? (10) ◎ 黑板上補充 H-H、D-D、C≡O 鍵長。 --

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