d (u ? v) = du ? dv, d (u?v) = udv vdu,

1
? ? ? ?

• ??? ????????
• 2. 1 ????????????
• 2. 2 ????
• 2. 3 ??????
• 2.4 ???

2
2.1????????????
(1)??
????

• ???????
• d (u ? v) du ? dv, d (u?v) udv vdu,
• df?(x) f?(x)d?(x) f?(x) ?(x)dx

3
?1 ? y x2 sinx, ? dy

• dy x2 d(sinx) sinx dx2
• dy x2 cosx dx 2x sinx dx

4
????

• ?? dz ???,fx(x,y) ???.
• ?2??? z x2y y2 ????.
• dz 2xydx (x2 2y)dy.

5
????

• ??????
• An(x) y(n) An-1(x) y(n-1) A0(x) y
  g(x)
• ? g(x) 0, ????????????
• y?? P(x)y? Q(x)y 0 (2.1)

6
????y1?y2???(2.1)??????,? ??????? y c1 y1
c2 y2
(2.2) ??????. ?????????(The linear homogenerous
second-order differential equation with constant
coefficients) y?? p y? q y 0
(2.3)
7
????(auxiliary equation)
?(2.3)???? y esx,Why? ?????
(2.4) (2.4)?????(auxiliary
equation).????? (2.4),???(2.3)?????
(2.5)
8
2.2 ????
???m??????(V 0)???????????Schroedinger??
(2.6)
?????
?
(2.7)
9
??A?????, ?????(?x??, ????????)???(2.7)???

(i) Ex ?????,? 0?? ????,???????????????????
(ii) ???x?????????????, ?,
x?????????
10
2.3 ??????
1 ??????
11
???I?III,Schroedinger???

• ??, ?I 0, ?II 0. (2.8)

12
???II, V0, Schroedinger???
(2.9)
??????
,
(2.10)

• ??E T V T, ????

13
????(2.5)??
(2.11)
?
(2.12)
??(1.10)??
14
????? x 0, l, ?II ?I ?III 0. ?
(i) x 0 ? A 0
(2.13)
(ii) x l
(2.14)
(2.14)??B?0,
??,
(2.15)
??n???? (Why? n0, E ? 0, ?II ? 0 ).
15
??(2.15)???
(2.15)
, n 1, 2, 3,
??i)???????,????n??ii) ????? iii) ???l??????
????(delocalization effect ).
16
???
(2.15) ??(2.13) ?
, n 1, 2, 3, (1.16)
??,n????????,n???????B??????????
17
,?
?? 2sin2t 1-cos2t, ?
, n 1, 2, 3, (2.17)
18
????????
19
??????

• i)      ??? n 1. ?n????,??????????????,???????
  ,?????????? Bohr correspondence principle.
• ii)      ?????(orthonormality).?

,
(2.18)
20
Exercise. ???????? ?????????? (2.18).
21
2 ??????
VV(x, y, z)V(x) V(y) V(z)
V(x, y, z) 0
(2.19)
V(x, y, z) ? ?abc??????
22
? ? ?(x, y, z) X (x) Y (y) Z (z) (????)
????Schroedinger ??,????????
(2.20)
23
??,??(2.20)????
(2.21a)
(2.21b)
(2.21c)
?????nx?ny?nz????
24
?????????
(3.22)
(2.21)
25
??????,(2.21)????
(2.22)
??(nx, ny, nz)(2, 1, 1), (1, 2, 1), (1, 1,
2)????????????, E 6h2/8ma2.
????????(degenerate state).
26
2.4 ??? (The Harmonic Oscillator)

• ??????????????????? (1/2)kx2, k?????
• ??????Hamilton??

(2.25)
Schroedinger ??
(2.26a)
27
(2.26b)
?
(2.27)
(2.28)
?????????????(Power-series solution)
28
?????????
(2.29)
(2.30)
??????? ???(Zero-point energy)
(1/2)h?
29
2. Hermite ??? H0(z) 1 H1(z) 2z
H2(z) 4z2 - 2 H3(z) 8z3 - 12z H4(z) 16z4
48z2 12 Hermite ???????? Hn 2zHn-1
2(n-1) Hn-2 (2.31)
30
3. ????? (Vibration of Molecules) ?????
????(reduced mass) ? m1m2 / (m1m2) ?? x
? R Re. ??? k d2V(x)/dx2, ? k d2U(R) /
dR2RRe.
U(R) ????,V(x)???U(R)??????
(2.32)
31
??????????? ?n ?1 ?????light (E2 E1) / h
(n2 n1)?e ?e
?????
????? ???3N, ??3,??3(?????),
2(????),??3N-6(?????) 3N-5(?????)
?????
32

• ?????????,?????????????????