課程名稱︰物理化學二
課程性質︰化學系必修
課程教師︰陳振中
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰98.01.15
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Section A (80%)
1. (20%) The Balmer Series has an equation of the following form:
1/λ = R [ 1 / 2^2 - 1 / n^2 ]
The energy expression of the Bohr's model of a one-electron atom:
En = -2.178*10^-18 Z^2 / n^2
(a) Derive the value of R in terms of the Planck's constant and light
velocity.
(b) What is the shortest wavelength of the emission spectrum of Be3+?
2. (20%) In a rather crude bonding model of the hydrogen molecular ion, H2 +,
the electron may be considered as a particle in a one-dimentional box whose
width is the same as the intermolecular separation r. Consider the ground
state only in the following questions. The Bohr radius
(hbar^2 * 4πε0 / m e^2 ) is equal to 0.53 angstrom; the H atom
ground-state energy (- m e^4 / 8 h^2 ε0^2 ) is -13.6 eV.
a. Express the kinetic energy T of the election as a function r.
b. Assuming the average potential energy V of the electron to be that of an
election at rest midway between the nuclei, express the total energy E
as a function of r.
c. Calculate the equilibrium bond distance r0 in angstrom and the
ground-state energy E0 in eV. The experimentl values are 1.06 angstrom
and -16.3 eV, respectively. [Hints: Your calculated ground state energy
should be higher than the experimental value.]
3. (20%) Cyclobutadiene, C4H4, is a four-carbon ring. Write the secular
equation for the π molecular orbitals of this planar molecule. Find the
energies of the orbitals. Predict the total π electronic energy of this
compound. Is these extra π electron stabilization (as butadiene or benzene
) in this molecule?
4. (20%) Consider the one-particle, one dimentional system with
potential-energy
V = V0 for l/4 < x < 3l/4, where V0 = hbar^2 / m l^2
V = 0 for 0 < x < l/4 and 3l/4 < x <l
V = ∞ elsewhere
Treat the system as a perturbed particle in a box. Find the first-order
energy correction E^(1) for the energy eigenstate with quantum number n.
Section B (20%)
＿
1. (10%) Consider the trial function of ψ(x) = √30 x (a-x) / a^(5/2). Use
the variational method to obtaion an upper bound to the ground-state energy
of a particle in-a-line of length a and compare the result with the true
value.
2. (10%) Consider a dimer formed by two monomers, viz. A and B. Assume that
each of the monomers has two energy states only. The normalized ground
states of the two monomers are designated as ψA0 and ψB0, whereas the
normalized excited states are denoted as ψA1 and ψB1, respectively. That
is, we have
^ ^
H ψ = E ψ and H ψ = E ψ
A A0 0 A0 B B0 0 B0
^ ^
H ψ = E ψ and H ψ = E ψ
A A1 1 A1 B B1 1 B1
(a) Suppose that the Hamiltonian of the dimer has the form HA + HB. The
first excited state of the dimer will contain one monomer in state 1 and
one in state 0. What is the degeneracy of the first excited state?
Calculate the energy difference between the ground state and the first
excited state of the dimer.
(b) Assume that the transition dipole moment of the monomer is written as
2 2
|< ψ | μ | ψ >| = |< ψ | μ | ψ >| = D .
A0 A A1 B0 B B1 0
Consider the transition between the ground state and the first excited
state of the dimer, calculated the absorption intensity in terms of D0.
(c) For the case of two interacting monomers, the Hamiltonian can be
written as
^ ^ -3 -5
H + H + V, where V = (μ ‧μ ) R - 3 (μ‧ R ) ( R ‧μ ) R .
A B A B AB A AB AB B AB
Write down the wavefunctions of the first excited states which are
stationary.
(d) The so-called exciton splitting refers to the frequency splitting
observed for interacting monomers, whereas such splitting is absent for
two noninteracting monomers. Explain the origin of exciton splitting.
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