推 trees880098 : 收化學系 05/25 23:34
課程名稱︰物理化學二-量子化學
課程性質︰必修
課程教師︰陳逸聰
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰104/4/21
考試時限（分鐘）：110分鐘
試題 :
註：約化普朗克常數(reduced Planck constant) 以 h 表示
Physical Chemistry (Ⅱ)--quantum Chemistry
1. (5%) In an X-ray photoelectron experiment, a photon of wavelength 150 pm
ejects an electron from the inner shell of an atom and it emerges with a
speed of 21.4 Mm s^-1. Calculate the binding energy of the electron.
2. (3%) (A) An unnormalized wavefunction for an electron in a carbon nanotube
of length L is sin(2πx/L). Normalize this wavefunction.
(4%) (B) For the system described in (A), what is the probability of finding
the electron in the range dx at x = L/2?
(4%) (C) For the system described in (A), what is the probability of finding
the electron between x = L/4 and x = L/2?
(4%) (D) Compute the expectation value of the kinetic energy of the electron
in the carbon nanotube.
h^2 d^2
3. (5%) (A) Confirm that the kinetic energy operator, -(----)----is hermitian.
2m dx^2
(5%) (B) The operator corresponding to the angular momentum of a particle is
h d
(---)--- , where is an angle. Is this operator hermitian?
i dψ
4. The wavefunction of an electron in a linear accelerator is
ikx ikx
Ψ = [cos(Χ)]e + [sin(Χ)]e , where Χ(chi) is a parameter. What is
the probability that the electron will be found with a linear momentum
(A) +kh ? (2%)
(B) -kh ? (2%)
(C) What form would the wavefunction have if it were 90 % certain that the
electron had linear momentum +kh? (2%)
(D) Evaluate the kinetic energy of the electron. (4%)
5. (10%) Determine the linear momentum and kinetic energy of a free electron
ikx
described by the wavefunction e with k = 3 nm^-1.
κL -κL 2
(e - e ) -1
6. (8%) Given the transmission probability T = {1 + ───────}
16ε(1 - ε)
and ε = E/V for a rectangular potential barrier.
Suppose that the junction between two semiconductors can be represented by
a barrier of height 2.0 eV and length 100 pm. Calculate the transmission
probability of an electron with energy 1.5 eV.
7. When β-carotene is oxidized in vivo, it breaks in half and forms two
molecules of retinal (vitamin A), which is a precursor to the pigment in the
retina responsible for vision.
http://ppt.cc/t7B4
The conjugated system of retinal consists of 11 C atoms and one O atom. In
theground state of retinal, each level up to n = 6 is occupied by two
electrons. Assuming an average internuclear distance of 140 pm, calculate
(3%) (A) the separation in energy between the ground state and the first
excited state in which one electron occupies the state with n = 7,
(3%) (B) the frequency of the radiation required to produce a transition
between these two states.
(3%) (C) Using your results, choose among the words in parentheses to
generatea rule for the prediction of frequency shifts in the absorption
spectra of linear polyenes:
The absorption spectrum of a linear polyene shifts to (higher/lower)
frequency as the number of conjugated atoms (increases/decreases).
-gx^2
8. (10%) Confirm that a function of the form e is a solution of the
Schrodinger equation for the ground state of a harmonic oscillator and find
an expression for g in terms of the mass and force constant of the
oscillator.
9. (3%) (A) Write the Schrodinger equation of a particle on a ring.
(8%) (B) Solve the Schrodinger equation of a particle on a ring to yield
1 im_lφ
wavefunctions of Φ (φ) = ──── e and energy levels of
2 2 m_l √(2π)
m_l h
E_m = ──── (m_l = 0, ±1, ±2, ±3, ...)
2I
(3%) (C) The particle-on-a-ring model is a crude but illustrative model of
cyclic, conjugated molecular systems. Treat the electrons in benzene as
particles freely moving over a circular ring of carbon atoms and calculate
the minimum energy required for the excitation of a electron. The carbon-
carbon bond length in benzene is 140 pm.
(3%) (D) Use the particle-on-a-ring model to calculate the minimum energy
required for the excitation of a electron in coronene, C24H12. Assume that
the radius of the ring is three times the carbon-carbon bond length in
benzene and that the electrons are confined to the periphery of the
molecule.
http://ppt.cc/xCF0
10.Terminology, description, and explanation.
(A) Wave-Particle duality and de Broglie relation. (4%)
(B) Uncertainty principle. (4%)
(C) Correspondence principle. (4%)
(D) Give two examples for a system that has a zero-point-energy at its
lowest quantum state. (4%)
http://ppt.cc/D0zp
http://ppt.cc/SOYi
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.77.19
※ 文章網址: https://www.ptt.cc/bbs/NTU-Exam/M.1430569318.A.291.html
※ 編輯: NTUkobe (140.112.77.19), 05/02/2015 23:19:37