課程名稱︰化學數學
課程性質︰化學系系必修
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）︰ 98 / 01 / 13
考試時限（分鐘）： 10:20 ~ 13:00，共160分鐘
是否需發放獎勵金：是
試題 :
1. Find a power series solution in powers of x. (10%)
y" - y' + xy = 0
k
2. The associated Legendre functions P (x) play a role in quantum physics.
n
k d^k Pn
They are defined by P (x) = (1-x^2)^(k/2)* ────
n dx^k
k^2
and are solutions of the ODE. (1-x^2)y" - 2xy' + [n(n+1)-────]y = 0
1-x^2
1 2 3
Find P (x), P (x), P (x) and verify that they satisfy. (9%)
3 3 3
3. Using the indicated substitutions, find a general solution in terms of Jν
and J-ν or indicate when this is not possible. (This is a sample of
various ODEs reduative to Bessel's equation.) (16%)
x^2 y" + xy' + 4 (x^4 - υ^2)y = 0 (x^2 = z)
4. The wavefunctions for a particle in a one-dimention box with potential of
╭ ∞ , x ≦ 0 , x ≧ l
V(x)= ┤
╰ 0 , 0 ＜ x ＜ l ∞ ∞
＿＿＿ │ │
╭ ψn(x) = √(2/l) sin (nπx/l) , 0 ＜ x ＜ l │ │
are ┤ ┴──┴→x
╰ ψn(x) = 0 , x ≦ 0 , x ≧ l 0 l
Prove that the wavefunctions of different quantum states are orthogonal to
∞
each other. That is ∫ ψm(x) ψn(x) dx = 0 , when m≠n . (10%)
-∞
( Hint: sinαsinβ = 1/2 [ cos(α-β) - cos(α-β)] )
5. Find the inverse matrix and prove your answer. (6%)
┌ ┐
│ 1 -2 -9│
│-2 -4 19│
│ 0 -1 2│
└ ┘
6. Find the directional derivative of f = x^2 + y^2 + z^2 at P: ( 2, -2, 1 )
→
in the direction of ａ = [ -1, -1, 0 ]. (5%)
7. Find the angle between the planes 4x + 3y - z = 2 and x + y + z = 1. (5%)
8. Find the velocity, speed, and acceleration of the motion given by
→
ｒ(t) = [ 5cost, sint, 2t ]
at point P: [ 5/√2 , 1/√2 ,π/2 ]. What kind of the curve is the path?
(12%)
9. What's the volume of the tetrahedron with verticles. (8%)
( 0 , 0 , 0 ) , ( 1 , 2 , 0 ) , ( 3 , -3 , 0 ) , ( 1 , 1 , 5 )
→ →
10. Let ｖ = [ y , z , 4z-x ], ｗ = [ y^2 , z^2 , x^2 ]. Find
→ →
(a) div (ｖ ×ｗ) (6%)
→ → → →
(b) curl (ｖ ×ｗ) + curl (ｖ ×ｗ) (6%)
11. Let Ψ(x,y,z) and Φ(x,y,z) be scalar fiends.
→ → →
Prove that ▽‧（▽Ψ ×▽Φ） = 0 (7%)
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