課程名稱︰化學數學
課程性質︰大二必修
課程教師︰陸駿逸
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2012.01.10
考試時限（分鐘）：10:20~12:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
整理書櫃翻到的 看了一下沒人po~
試題 :
Use enough words(in Chinese or English) to explain your reasoning and thought
1. (25 pts)
(a) Use the method of separation variable, to reduce the following PDE into
two ODEs
2 1 2 1 δ δ 1 δ^2
▽ f(x,y)= ----------f(x,y) ,where ▽= --- ---r--- + --- ----
x^2 + y^2 r δr δr r^2 δθ^2
(註:δ為偏微分符號)
(b) Try to solve these ODEs to find some solutions of the original PDE.
Obtain as many solution as you can.(Hint: you can try the power law r^a
for the radius function.)
2. (10 pts)
Let f(x)=2x for -10<x<10. Also, f(x+20)=f(x). Plot the function, and find
the Fourier coefficients for the sine and cosine a1, a2, a3, b1, b2, b3.
3. (10 pts)
Plot the function g*f, where g(x)-exp(-25x^2)+0.5exp[-25(x+0.3)^2], and
f(x)= Σδ(x-n)
n=3,4,..6
4. (15pts)
Find outt all the symmetry operations of the following substituted
cyclobutane. X
_____|
∕___∕
|
X
5. (50 pts)
Cosider the subgroup C of the molecular BF . One can construct a
3v 3
representation by the six p orbitals as shown below.
φ
6 φ C | E 2C 3σ
○ 5 φ 3v | 3 v
○F○ 4 ----------------------------
○ ╲ ○ ╭╮ A1 | 1 1 1
B—○F○ φ ˇ│ ----------------------------
φ ╱ ○ 3 ╰╯ A2 | 1 1 -1
8 ○ ----------------------------
○F○ φ E | 2 -1 0
○ 7
(a) Draw the six functions Eφ, C φ , ....
3 3 3
(b) Express the above (six) function in terms of the basis{φ φ φ φ φ φ}
3 4 5 6 7 8
(c) Starting from φ, use the A1 character to find out (project) the LCAO
3
which forms the A1 representation.
(d) Starting from φ, use the E character to find out (project) the LCAO,
3
called χ, which belongs to the E representation. Draw a figure forχ.
1 1
(e) Suppose that {χ,χ} forms the above E representation. Draw a figure
1 2
for C χ. Use the fact that C χ should be the linear combination of
3 1 3 1
χ and χ. Assume that χ,χ are orthogonal, findχ.
1 2 1 2 2
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