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）︰2013.5.2
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 :
1. Show that the function (r/a )exp(-r/2a ) is a solution of the differential
0 0
equation for l=1.
^^
(英文)
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?(24)
2. Calculate the mean value of the radius <r> at which you would find the 1 s
+ +2
electron in H-atom, He , and Li . The wave function is
Ψ(r)=(1/π)^(1/2) (〖Z/a 〗^(3/2) exp(r/a) )
0 0
If the ionization energy for H-atom is 13.6 eV, calculate the ionization
+ +2
energies of He and Li . (20)
3 3 5 5 7 7
3. Why Φ(x)=(x/a-x / a ) + α(x / a - x / a ) is not a trial wave function
for a particle in a box? Please modify to to give a valid wave function.
(10)
4. Why the Schrodiger equation cannot be solved analytically for He? Describe
how to use approximate method step by step ( problem, model, experiments,
and importance) to solve the Schrodinger equation for many-electron atoms.
(20)
5. In the lecture, we have used variation method to obtain the minimum energy
2
of He as <H>=(2Z - 27/4 Z )‧(13.6eV), calculate the first ionization
e e
energy of He?(12)
6. (a) Establish a group which is generated by the following elements:(24)
│1/2 [(3)^1/2]/2 │ │-1/2 -[(3)^1/2]/2 │
A =│ │ B =│ │
│[(3)^1/2]/2 -1/2 │ │[(3)^1/2]/2 -1/2 │
(b) Show that each of the four properties which define a group is
satisfied by your list of operations.
(c) What is the order of this group? Is this group Abelian? How many
classes?
7. Formaldehyde (H CO) belongs to the C group.(24)
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 reducigle 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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