課程名稱︰化學數學 課程性質︰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）︰100/11/08 考試時限（分鐘）：110min (延長為130min) 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Given a hermitian matrix A as below ┌ ┐ │ 1 1-i │ A= │ │ │ 1+i 2 │ └ ┘ (a) Calculate its normalized eigenvectors│v1>,│v2> and the eigenvalues a1,a2 (b) Calculate the matrices│v1><v1│and│v2><v2│ (c) Show that│v1><v1│+│v2><v2│=1 (d) Show that a1│v1><v1│+a2│v2><v2│=A (e) An arbitrary vector│u>=┌ ┐ can be expressed as u1│v1>+u2│v2>, │ 2i │ │ │ │1+i │ └ ┘ calculate u1,u2. 2. Prove that the eigenvalues of a hermitian matrix are always real. 3. In the H atom energy level problem, we can get the 3d orbitals. -r -r -r ψxy=xye ψyz=yze ψxz=xze 2 2 2 2 -r 2 2 2 2 -r 2 2 2 2 -r ψ(x -y )=(x -y )e ψ(y -z )=(y -z )e ψ(x -z )=(x -z )e (上面的=都是"正比於") 2 2 2 where r=√(x +y +z ) Choose 5 linear independent orbitals, and use the Gram-Schmidt process to construct an orthogonal basis. You may need the following integrals: ∞ ∞ ∞ 2 2 -2r ∫ ∫ ∫ x y e dxdydz=3π/2 -∞ -∞ -∞ ∞ ∞ ∞ 4 -2r ∫ ∫ ∫ x e dxdydz=9π/2 -∞ -∞ -∞ 2 4. Consider the vector space of polynomial with power less than 5. Take{1,x,x , 3 4 x ,x } as a basis, construct the matrices correspond to the linear operators ^ d ^ d A = x ── B = ── dx dx 2 ^ d ^^ ^ d d ^^ C = x ── = AB D = ── x ── = BA 2 dx dx dx ^ iA 5. Let A=(√3)σx +σy where σx =┌ ┐ σy =┌ ┐ Calculate e │0 1│ │0 i│ │ │ │ │ │1 0│ │-i 0│ └ ┘ └ ┘ 6. Solve the coupled linear ODE x'(t)=2y(t)+x(t), y'(t)=2x(t)+4y(t) with the initial conditions x(0)=1, y(0)=2. --

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