作者leehoni (leehoni)
看板NTU-Exam
标题[试题] 96下 陈义裕 量子物理下 期末考-2
时间Thu Jun 12 23:08:39 2008
课程名称︰量子物理下
课程性质︰必修
课程教师︰陈义裕
开课学院:理学院
开课系所︰物理系
考试日期(年月日)︰97/6/11
考试时限(分钟):120分钟
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试题 :
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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