課程名稱︰量子物理下 課程性質︰必修 課程教師︰陳義裕 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰97/6/11 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 謝謝 （如未明確表示，則不予發放） 試題 : 3.(30points) (a)(5points)Please use the so-called tight binding model for crystalline solids to explain how energy bands are formed in one-dimentional crystals. (b)(10points)Please use the band theory to explain the electrical conductivity of insulators,conductors, intrinsic semiconductors. (c)(5points)Drude's modelfor the electrical conductivity of metals predicts that the conductiity σis given by n(q^2) σ= ────τ (1) m whereτis the relaxation time,and n,q and m refer to the number density, the charge,and the mass of the charge carries responsible for the electrical conduction . Please give a classical derivation of this result. (d)(5points)Based on Eq(1),classical physics(mistakenly)predictes that the temperature dependence of σis 1 σ～── √T Please repeat this (erroneous) reasoning. (e)(5points) Based on Eq(1), quantum mechanically one can argue that actually 1 σ～── T Please explasin how the argument goes. 4.(20points)Let S Px Py Pz be the s-orbital and the three p-orbitals, respectiviely: 1 S = √── 4π, 3 Px = √──sinθcosψ 4π , 3 Py = √──sinθsinψ 4π , 3 Pz = √──cosθ 4π , which are expressed in the standard spherical coordinates. They are orthonormal orbitals under the inner product defined by 2ππ ＜f∣g＞ =∫ ∫ f g sinθdθdψ 0 0 We will use then to cinstruct orbitals suitable for the sp2 hybridization scheme. (a)(3points)Please verify explicitly that Px is simply a Pz orbital re-oriented in the x-axis.(A similar remark may be made about the Py orbital.) (b)(3points)Please verify explicitly that Pxcosα+Pysinα is simply a orbital oriented in a direction in the x-y plane making an angle α with the x-axis. (c)(3points) Write Ψ1 = aS + √(1-a^2)Px for some number a, with 0 < a < 1. Please explain why this wavefunction represents a "lobe" of probability pointiong in the positive x direction. (d)(3points)Combining the results of (b)and(c), please give a physical description of what the following wavefunctions represents. 1 √3 Ψ2 = aS + √(1-a^2)(-—Px+ —Py) 2 2 1 √3 Ψ3 = aS + √(1-a^2)(-—Px- —Py) 2 2 (e)(3points)Please find the value of "a" such that Ψ1,Ψ2,Ψ3 are mutually orthogonal wavefunctions. (f)(5points)What does all the above have to do with sp2 hybridization and the molecular structure of BF3 (boron trifluoride)? (Note: Boron and fluorine are located in the 3rd and the 7th column of the periodic table, repectivity.) --

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