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 - ν^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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