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 :
1.(25points)It is known that the average thermal energy for a normal mode
of a crystal as the temperature T can be taken as
hω
ε = ────── h:Planck constant/2π
exp(βhω)-1
where ω is the (angular) frequency of the normal mode, and β=1/(kbT),
kb:Boltzmann constant
We will consider Debye's model of specific heat for a linear chain
consisted of N identical atoms. However, we assune that, corresponding
to a given wavenumber k, there are three types of allowed vibrational
motion of the crystal lattice : two transverse and one longitudinal.
For simplicity, we assume that the dispersion relation satisfies
ω=c∣k∣for all the normal modes having a wavenumber k with c
signifying the wave speed.
(a)(5points)Using periodic boundry conditions, please explain why it is
natural that k=2πj/Na, where j is an integer ranging from
-N/2 to N/2.
(b)(5points)Please show that the total thermal energy of crystal is
3Nkb(T^2) Θ/T x
E = ──────∫ ── dx
Θ 0 exp(x)-1
(c)(5points)For T very large, please show that the specific heat per atom
is approxiamtion 3kb.
(d)(5points)For T very small, how does the specific heat depend on the
temperature T?
(e)(5points)If we generalize this model to a two-dimensional lattice,
keeping the same three types of vibration and the same dispersion
relation for all the normal modes, then what is the temperature
dependence of the specific heat as low T?
2.(25points)
(a)(10points)Please argue why the relative magnitude of the elcectronic
energy, the molecular vibrational energy,and the molecular rotational
energy is approximately of the ratio
m m
1:√ ──:──
M M
where m and M are the mass of electron and the nucleus, respectively
.Then please utilize this fact to give a brief account of the so-called
Born-Oppenheimer approximation of the structure of molecules.
(b)(8points)Consider two hypothetical homonuclear diatomic molecules A2
and B2. It is known that both the atomic nu,ber and the mass number for
Nucleus A and Nucleus B are the same. The only known difference
between A and B is that A is a fermion while B is a boson. Please
explain why A2 and B2 molecular are not expected to have the same
vibration and rotation molechlar spectra.
(c)(7points)Please explain how one may use the experimental data of the
dissociation energy of molecules to indirectly prove the zero-point
energy of a simple harmonic oxcillator is 1/2 hω.
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※ 編輯: leehoni 來自: 218.168.68.56 (06/12 22:32)
※ 編輯: leehoni 來自: 218.168.75.246 (06/13 17:26)
> -------------------------------------------------------------------------- <
作者: leehoni (leehoni) 看板: NTU-Exam
標題: [試題] 96下 陳義裕 量子物理下 期末考-2
時間: Thu Jun 12 23:08:39 2008
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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