[試題] 100上 陸駿逸 化學數學 期中考

[twc94 (蔡哲)]

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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