[試題] 101下 張大釗 物理化學下 第三次小考

[perhdball (帥德)]

課程名稱︰物理化學下
課程性質︰農化系必修
課程教師︰張大釗
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2013.5.23
考試時限（分鐘）：50分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :
                                              2
1. Under what condition, the function exp(-βx ) satidfies the Schrodinger

                              2     2    2
   equation, [-(h/2π)^2/2u](d Ψ/dx )+kx /2 Ψ(x)=EΨ(x), for the quantum

   harmonic oscillator, what is E?(12)




2. What are the average values of the kinetic and potential energies for a

   quantum harmonic oscillator of the ground state,

               1/4        2
   Ψ(x)=(α/π)  exp(-αx /2) .  (12)



3. In the lecture, we have used variation method to obtain the minimum

                          2
   energy of He as <H>=(2Z - 27/4 Z )‧(13.6eV) , calculate the first
                          e        e

   ionization energy of He?(10)




4. Show that the function (r/a ) exp(-r/2a ) is a solution of the differenrial
                              0           0

   equation for l=1.


        2                                    2             2
       h          d      2  dR(r)           h l(l+1)      e
   -  ───── ── [ r ───── ] + [ ───── - ──── ] R(r)=E R(r)
      2m r^2      dr          dr             2m r^2      4πεr
        e                                      e             0

    What is the eigenvalue? What is the n value for this function?(12)





                        35                -12
5. The bond length for H  Cl is 127.5 X 10    m. Where does the axis of

   rotation intersect the molecular axis and calculate the moment of inertia?

   (10)



                                             2          2
6. Based on the equation, [-(h/2π)^2/2ur ][d Φ(ψ)/dψ ]=EΦ(ψ),
                                         0

   what is the eigenfunction Φ(ψ) normally given in the lecture?

   Using this two-dimensional polar coordinates to calculate E and draw a

   rotation spectrum of HCl.(24)




7. Draw a picture showing all angular momentum sones with l=2. Calculate the

                  2   2
   magnitude of (l  +l  ) for l=2 and m =2. (12)
                  x   y                l


8. In the lecture, we have discussed the stretching vibrations of HCN, the

                               2              2
   potential energy U=k (δr  ) /2 + k (δr  ) /2 , and the kinetic energy
                       1    12        2    23
        .        .        .
   T= m z /2 + m z /2 + m z /2  , list the major steps to convert them to the
       1 1      2 2      3 3

   normal modes and obtain:

   .         (m2+m3) .     m3  .
   z  =  - ─────Q  - ── Q
    1          M      1    M    2

   .     m1 .      m3  .
   z  = ──Q  -  ── Q
    2    M   1     M    2

   .     m1 .     (m1+m2)  .
   z  = ──Q1 + ─────Q
    3    M          M       2              o      o
                                          r      r
                                           12     23
                                        ●──●──●

                                    ──┼──┼──┼──
                                         z     z     z
                                          1     2     3




  (20)

9. Formaldehyde (H CO) belongs to the C  group.(18)
                  2                    2v

             E      C     σ    σ'
                     2     v     v
         ─────────────────────────
                                              2   2   2
         A   1      1      1     1    z      x , y , z
          1

         A   1      1     -1     -1   R      xy
          2                            z

         B   1     -1      1     -1   x, R   xz
          1                               y

         B   1     -1     -1     1    y, R   yz
          2                               x

(a) List the reducible representation of this molecule.

(b) How many vibrational modes does formaldehyde have? How many of them are

    in-plane motion?

(c) Use irreducible representation to list the infrared active modes.

(d) Use irreducible representation to list the Raman active modes.


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