PPP - Beijing Normal Universityphysics.bnu.edu.cn/application/faculty/guowenan/QM/... ·...
Transcript of PPP - Beijing Normal Universityphysics.bnu.edu.cn/application/faculty/guowenan/QM/... ·...
2.3. �Ù⇣/P 27
2.3 ���ÙÙÙ⇣⇣⇣///PPP
⇤Q�*9'�(π(œm. ø˝V (x) = 12kx
2. k/9'˚p⇥íP◊õ
f = �kx
.
œxõf⇢
�kx = mx (2.68)
„:
x = A sin(!t+ �) (2.69)
p = mx = m!A cos(!t+ �) (2.70)
v-! =p
k/m:Í6ëá�ø˝ÔÂô:V (x) = 12m!
2x2. A1˝œE≥ö⇢ 1
2kA2 = E. ¯M�1�ÀM
n≥ö⇥
⌘Ï_ÔÂ(»∆�õfeô–®π↵, H = p2
2m + 12m!
2x2
x =@H
@p
p = �@H@x
(2.71)
M⇧1/®œÑöI✏��⇧/[�π↵⇥
œPõfœ⇣/P⇢
i~@ (x, t)@t
= H (x, t) (2.72)
H⇤Qö�≈µ⇢ (x, t) = E(x)e�iEt/~. zÙË⌃·≥ö�S-eq
H E(x) = E E(x) (2.73)
s
� ~22m
d2 E
dx2+
1
2m!
2x2 E = E E (2.74)
$πdÂ~!/2,
� E
00
m!
~+
m!
~ x2 E =
2E
~! E (2.75)
⌃ê⇢[m!~ ] = 1L2 . $πœ≤¯�⇥
öIm!/~ = ↵2, x0 ⌘ 1/↵/�¶�Í6Ñ�¶UM�⇡x/(1/↵) ⌘ ⇠, /ÂÍ6�¶UM¶œÑ�¶.
12~!÷:Í6Ñ˝œUM, £H� ⌘ E/( 12~!)1/ÂÍ6˝œUM¶œÑ˝œ.
�d2 E
d⇠2+ ⇠
2 E = � E (2.76)
ñHvÅP⇠ ! ±1↵⌦bπ↵Ñ⇣—L:. dˆÔ½eâ� E , π↵�:
d2 E
d⇠2� ⇠
2 E = 0. (2.77)
„:
E / e±⇠2/2
. (2.78)
⇤Q0 E ! 0(_⇢� , ⌦✏Í÷‘-’˜⇥
28 CHAPTER 2. �ÙÓò
(^‡w‹⌅�⌘Ïæ E = u(⇠)e�⇠2/2, &e(2.76)
d2u
d⇠2� 2⇠
du
d⇠+ (�� 1)u = 0 (2.79)
d:W�ÑHermiteπ↵⇥
¬flÔó⇢
u = c0, if � = 1, 1/E = 12~! (2.80)
u = ⇠, if � = 3, 1/E = (1 + 12 )~! (2.81)
u = 2⇠2 � 1 if � = 5, 1/E = (2 + 12 )~! (2.82)
Ÿ1/Ñ∆π↵ÑM‡*„�
�,0, Gæ‡wßp„
u =1X
k=0
ck⇠k. (2.83)
_⇢�å˘ø˝�¸Ù„ á. ⌘Ï⌃+⇤Q⇢
1. vá„: ue = c0 + c2⇠2 + c4⇠
4 + · · ·
2. Gá„: uo = c1⇠ + c3⇠3 + c5⇠
5 · · ·
⌃(2.83)&e(2.79),
X
k=2
ckk(k � 1)⇠k�2 � 2⇠X
k=1
ckk⇠k�1 + (�� 1)
X
k=0
ck⇠k = 0 (2.84)
9ôK�©k˝Œˆ�Àpw
X
k=0
ck+2(k + 2)(k + 1)⇠k � 2X
k=0
ckk⇠k + (�� 1)
X
k=0
ck⇠k = 0 (2.85)
‘É�!B
ck+2 =2k + 1� �
(k + 2)(k + 1)ck (2.86)
ÇúÂSc0,ÔÂó˙c2, c4, c6, · · · . ÂSc1, ÔÂÂSc3, c5, · · · .F/ŸÃœ@�*Óò�⌘Ï⇢«vá„eÙ�. dˆk = 2m:vp. (mà'ˆ�
c2m+2
c2m! 1
m+ 1. (2.87)
S⇠ ! 1ˆ, ue-kä'ÑyäÕÅ. ‡d
ue ⇡X
m
(⇠2)m
m!= e
⇠2
. (2.88)
Ÿ¸Ù E = ue�⇠/2 —c�
:�MŸÕ≈µ—��⌘⌘⌘ÏÏÏ≈≈≈{{{ÅÅÅ©©©‡‡‡wwwßßßppp„„„---≠≠≠:::⇢⇢⇢yyy✏✏✏���
Çú� = 2n + 1, n:tp�£Hßp„1⇢»b:⇢y✏�Œ��M—c⇥Ÿ*⇢y✏1/W�
ÑHermite ⇢y✏Hn(⇠). Ôã, ƒö�ÿBÑ˚pcn = 2n. ‘Ç: H0(⇠) = 1, H1(⇠) = 2⇠, H2(⇠) =
c0 + c2⇠2 = 4⇠2 � 2.
Ÿ*�/ 12~!:UMÑ˝œ��⇧Ù
En = (n+ 1/2)~!. (2.89)
2.3. �Ù⇣/P 29
Figure 2.4: Ä⇣/PÑ˝œ,Å�.
˘îÑ‚˝p1/
n = NnHn(⇠)e�⇠2/2 = NnHn(↵x)e
� 12↵
2x2
. (2.90)
v-Nn:R��˚p:
Nn = (↵
2nn!p⇡)
12 . (2.91)
ŸÔÂ)(Ñ∆⇢y✏Ñ'(ó0Z 1
�1Hn(⇠)Hm(⇠)e�⇠
2
d⇠ =p⇡2nn!�m,n (2.92)
1d Z 1
1 m(x) n(x)dx = �m,n (2.93)
Ÿ1/‚˝pÑc§R�'⇥
�ÕÅÑ/˙�‚˝p
0 =↵
12
⇡1/4e� 1
2↵2x2
(2.94)
1/⌘ÏüâÑÿØ˝p⇥
,�¿—�
1(x) =(2↵)
12
⇡1/4↵xe
� 12↵
2x2
(2.95)
œxõfJ…⌘Ï�Ÿö˝œE, ÔÂó˙/PÑ/EA2 = 2E/(m!2). ˘é˙�˝œE = 1
2~!, π◆ó˙œx/EA = ~/(m!). ÔÂ↵0�ÍÍÍ666���¶¶¶UUUMMM111///˙���˘îîîÑÑÑA.
F/ œxõf�íPîÂ˙∞(±AKÙ⇥œPõfJ…⌘ÏŸ�/�Ñ⇥⌘Ï°óíP—˙Ÿ*⇤
Ùчá
p = 2
Z 1
1/↵| 0(x)|2dx =
2p⇡
Z 1
1e�⇠2d⇠ ⇡ 0.16 (2.96)
⌘Ïçe↵,�¿—� 1.
ñHeB�Ô‡Mn. )(d 21
dx = 0,
d(⇠2e�⇠2
)
d⇠= 0, (2.97)
ó⇠ = ±1, Ù�íP(±1/↵⌅˙∞‡á�'⇥
ÿÔÂBdˆÑœxÅ:⇢ 12m!
2A
2 = 32~!, ó0A =
p3/↵.
ÿ íP(Å:K�˙∞чá
p = 2
Z 1
A
21dx =
4p⇡
Z 1
p3⇠2e�⇠
2
d⇠ = 0.112 (2.98)