課程名稱︰物理化學二
課程性質︰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）︰98.12.19
考試時限（分鐘）：180
是否需發放獎勵金：yes
（如未明確表示，則不予發放）
試題 :
Section A (80%)
1.(20%) Suppose a system has the energy eigenfunction of Ψ= N exp(-bx^4),
where N and b are constants. Find the potential-energy function V(x)
for the system. Hint: note that the zero level of energy is arbitrary,
and can be chosen at our convenience.
2.(20%) If the normalized variation function ψ=(√3/L^3/2)x for 0≦x≦L is
applied to the particle-in-a-one-dimensional-box problem, calculate
the variational intergral. Is the variation principle obeyed? Why?
^ h ┌ 0 -i ┐
3.(10%) The operator of Sy = ─│ │has two eigenfunctions. (h是有一撇的)
2 └ i 0 ┘
1 ┌ 1 ┐
(a) One of the eigenfunction is ─ │ │.
√2 └ i ┘
Determine the corresponding eigenvalue.
^
(b) What is the second eigenfunction for Sy?
Determine the corresponding eigenvalue.
4.(30%) Consider a particle moving in a ring. (圖就是一個圓 半徑R 角度ψ)
(a) To ensure that the particle moves freely on the circumference of a
ring, how should we write down the potential energy?
(b) Can we treat the motion of the particle as a one-dimensional
problem?
(c) What is the Schrodinger equation of the particle in terms of ψ?
(d) Write down a possible solution for the Schrodinger equation.
(e) Determine the normalized wavefunctions.
(f) What are the allowed energies for this system?
Section B (20%)
1.(5%) Referring to the question 3 of Section A. Suppose an electron is in the
┌ α ┐
spin state│ │. If Sy is measured, what is the probability that the
└ β ┘
measured result is h/2 ? (h有一撇)
2.(5%) Two Hermitian operators anticommute:
^ ^ ^^ ^^
｛A,B｝= AB + BA = 0
^ ^
Is it possible to have a simultaneous eigenket of A and B ? Prove or
illustrate your assertion. [Hints: assume that all the eigenvalues of
^ ^
A and B are non-zero] (h有一撇)
^ ^
3.(10%) For a one-dimensional system, we have [x,Px] = ih and it can be shown
that the position opertor has the following well-defined matrix element:
d
〈p'│x│p'〉= ∫dx'｛ih ─〈p'│x'〉｝〈x'│p'〉. (h有一撇 後d是偏微)
dp'
As an analogy, one may suggest that the angular momentum and angle
^ ^
opertors would obey a commutator relation of [ψ,Lz] = ih. (h有一撇)
Explain why such expression would cause a serious conceptual problem.
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