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)