Сербо, Хриплович. Квантовая механика

¤¥©á⢨¥	®¯¥à â®à a^ -	á®áâ®ï-¨¥ jni ¯¥à¥¢®¤¨â ¥£® ¢	á®áâ®ï-¨¥
jn − 1 i, â®	¥áâì ¯®-¨¦ ¥â	í-¥à£¨î á®áâ®ï-¨ï - 1. •â®	¯®§¢®«ï¥â

¨á¯®«ì§®¢ âì 㤮¡-ãî ¨-â¥à¯à¥â æ¨î: á®áâ®ï-¨¥ jn i ᮤ¥à¦¨â n ®¤¨- ª®¢ëå ç áâ¨æ (ª¢ -⮢) á í-¥à£¨¥© E = 1 (¨«¨ h! ¢ ®¡ëç-ëå ¥¤¨-¨æ å) ª ¦¤ ï, ®¯¥à â®à a^+ - §ë¢ îâ ¯®¢ëè î騬 ®¯¥à â®à®¬ ¨«¨ ®¯¥à â®à®¬ ஦¤¥-¨ï â ª®© ç áâ¨æë, ®¯¥à â®à a^ | ¯®-¨¦ - î騬 ®¯¥à â®à®¬ ¨«¨ ®¯¥à â®à®¬ ã-¨ç⮦¥-¨ï. ‡ ¬¥â¨¬ ¥é¥, ç⮠ᮡá⢥--ë¥ §- ç¥-¨ï ®¯¥à â®à

n^ = a^+a^ = H^ − 1 2

à ¢-ë n, ¯®í⮬ã n^ - §ë¢ îâ ®¯¥à â®à®¬ ç¨á« ç áâ¨æ.

• ©¤¥¬ ª®íää¨æ¨¥-â cn. „«ï í⮣® ¢ëç¨á«¨¬ -®à¬ã ¢¥ªâ®à (7.2):

2

:

h nj a^a^+ jn i = h nj H^ + 1=2 jn i = n + 1 = cn

Žâáî¤ cn

= p

+ 1

n

. ’ ª¨¬ ®¡à §®¬, á®áâ®ï-¨¥ jn i ¬®¦¥â ¡ëâì § -

¯¨á -® â ª:

(^a+)n

n

i

=

p

0

i

;

j

n

j

®â«¨ç-ë¥ ®â -ã«ï ¬ âà¨ç-ë¥ í«¥¬¥-âë ®¯¥à â®à®¢ ஦¤¥-¨ï ¨ ã-¨ç⮦¥-¨ï à ¢-ë

h n	+ 1 ^+	jn i	=	^	+ 1	i	= p			:
									+ 1
	j a			h nj a jn			n

‚®«-®¢ ï äã-ªæ¨ï ®á-®¢-®£® á®áâ®ï-¨ï ¬®¦¥â ¡ëâì - ©¤¥- ¨§ ãá«®¢¨ï

a^ 0(x) = 0:

•â® ¤ ¥â

	e−x2=2
0(x) =	p4		:

„«ï ¢®«-®¢®© äã-ªæ¨¨ á n > 0 ¯®«ãç ¥¬ ª®¬¯ ªâ-®¥ ¢ëà ¦¥-¨¥

	(^a+)n e−x2=2
n(x) =	p				p4		:
		n	!

‚Ž••Ž‘›

7.1. • ©â¨ 'n(p) ¤«ï «¨-¥©-®£® ®á樫«ïâ®à .

21

7.2. „«ï «¨-¥©-®£® ®á樫«ïâ®à áà ¢-¨âì ª« áá¨ç¥áªãî dWª« áá=dx ¨ ª¢ -⮢ãî j n(x)j2 ¯«®â-®á⨠¢¥à®ïâ-®á⨠¯à¨ n = 0.

’® ¦¥ ¤«ï dWª« áá=dp ¨ j'0(p)j2.

•©â¨ ¢¥à®ïâ-®áâì ⮣®, çâ® ¢ ®á-®¢-®¬ á®áâ®ï-¨¨ ®á樫«ïâ®à

-室¨âáï ¢ ª« áá¨ç¥çª¨ -¥¤®áâã¯-®© ®¡« á⨠jxj > `.

7.3.• ©â¨ ¬ âà¨ç-ë¥ í«¥¬¥-âë xf i; pf i; (x2)f i ¤«ï «¨-¥©-®£® ®á樫«ïâ®à .

7.4.• ©â¨ x ¨ p ¤«ï «¨-¥©-®£® ®á樫«ïâ®à ¢ n-¬ á®áâ®ï-¨¨.

x8. ‚६¥--®¥ ãà ¢-¥-¨¥ ˜à¥¤¨-£¥à

‚ ª« áá¨ç¥áª®© ¬¥å -¨ª¥ ¨¬¯ã«ìá ¨ í-¥à£¨ï á¢ï§ -ë á ¤¥©á⢨¥¬ S(x; t) á®®â-®è¥-¨ï¬¨

p = @S@x ; E = −@S@t :

…᫨ ¢ ª¢ -⮢®© ¬¥å -¨ª¥

p ! p^ = −ih@x@ ;

â® ¥áâ¥á⢥--® ®¦¨¤ âì, çâ®

E ! ih@t@ :

•à®¢¥à¨¬, çâ® ¤«ï ¯«®áª®© ¢®«-ë

(x; t) = A ei(px−Et)=h

íâ® â ª ¨ ¥áâì: ih@ =@t = E .

Š®-¥ç-®, ¢áe íâ® «¨èì - ¢®¤ï騥 á®®¡à ¦¥-¨ï, ¯®ª §ë¢ î騥 ¥áâ¥á⢥--®áâì á«¥¤ãî饣® ã⢥ত¥-¨ï: ¢ ª¢ -⮢®© ¬¥å -¨ª¥ ¯®- áâ㫨àã¥âáï “˜ ¢ ¢¨¤¥

ih	@ (r; t)	= H^	(r; t) = 2	h2		+ U(r)3 (r; t) :
				2m
	@t		4−			5
…£® ᢮©á⢠:
1. “˜ «¨-¥©-®: ¥á«¨			1(r; t) ¨ 2(r; t) | à¥è¥-¨ï “˜, â® c1				1 +
c2 2 â ª¦¥ ï¥âáï à¥è¥-¨¥¬ “˜ (¯à¨-樯 á㯥௮§¨æ¨¨).
2. “˜ ¨¬¥¥â ¯¥à¢ë© ¯®à冷ª ¯® ¢à¥¬¥-¨, ¯®í⮬㠧- ç¥-¨ï							(r; t)

¢ «î¡®© ¬®¬¥-⠢६¥-¨ ¯®«-®áâìî ®¯à¥¤¥«ï¥âáï, ¥á«¨ ¨§¢¥áâ- (r; t0) ¢ -¥ª®â®àë© ¬®¬¥-⠢६¥-¨ t0.

22

„«ï áâ 樮- à-®£® à¥è¥-¨ï

(r; t) = n(r) e−iEnt=h

¯«®â-®áâì ¢¥à®ïâ-®á⨠j (r; t)j2 -¥ § ¢¨á¨â ®â t. Ž¡é¥¥ à¥è¥-¨¥ â - ª®¢®

		X
(r; t) =		cn e−iEnt=h	n(r) ;
£¤¥		n
	Z
cn =		n (r) (r; 0) d3r :

’ ª¨¬ ®¡à §®¬, í¢®«îæ¨ï (r; t) á â¥ç¥-¨¥¬ ¢à¥¬¥-¨ ®¯¨áë¢ ¥âáï ãà ¢-¥-¨¥¬

(r; t) = Z	G(r; r0; t)	(r0; 0) d3r0 ; G =				n	n(r) n(r0) e−iEnt=h :
						X
”ã-ªæ¨ï ƒà¨-	G(r; r0; t) 㤮¢«¥â¢®àï¥â ãà ¢-¥-¨î
		ih	@G		= H^ (r) G
			@t
á - ç «ì-ë¬ ãá«®¢¨¥¬
	G(r; r0; 0) =			n(r) n(r0) = (r				−	r0) :
		n
		X
ˆ§	hEi = Z	(r; t) H^ (r; t) d3r =					n	En jcnj2
							X
¢¨¤-®, çâ® cn	¥áâì ¬¯«¨âã¤			¢¥à®ïâ-®á⨠®¡- à㦨âì ã á¨á⥬ë

í-¥à£¨î En. • ¡®à ¢¥«¨ç¨- cn ¥áâì ¢®«-®¢ ï äã-ªæ¨ï á¨áâ¥¬ë ¢ í-¥à£¥â¨ç¥áª®¬ ¯à¥¤áâ ¢«¥-¨¨.

•«®â-®áâì ⮪

ˆ§¬¥-¥-¨¥ ¯«®â-®á⨠¢¥à®ïâ-®á⨠%(r; t) = j (r; t)j2 á® ¢à¥¬¥-¥¬ ®¯à¥- ¤¥«ï¥âáï ãà ¢-¥-¨¥¬

	@%		=		@	+	@	:
	@t				@t		@t

•®¤áâ ¢¨¢ @ = @t ¨§ “˜, ¯®«ã稬 ãà ¢-¥-¨¥ -¥¯à¥àë¢-®áâ¨

@%

=

ih

h r2

− r2 i = −rj ;

@t

2m

1

[

(

−

ih

) + ( ih

) ] :

j =

2m

r

− r

23

„«ï

= p% ei

¨¬¥¥¬

j = %hrm :

‚ ç áâ-®áâ¨, ¤«ï ¯«®áª®© ¢®«-ë = A ei(kr−!t) ¯«®â-®áâì ⮪ à ¢-

j = jAj2 v ; £¤¥ v =	hk		:
	m

x9. Ž¤-®¬¥à-®¥ à áá¥ï-¨¥

„«ï ¯®â¥-æ¨ «ì-®© í-¥à£¨¨ 㪠§ --®£® - à¨á. 2 ¢¨¤ (U (x) ! 0 ¯à¨ x ! −1, U (x) ! V ¯à¨ x ! +1) § ¤ ç à áá¥ï-¨ï ¯à¨ E > V

•¨á. 2: •®â¥-æ¨ «ì- ï í-¥à£¨ï ¤«ï á«ãç ï ®¤-®¬¥à-®£® à áá¥ï-¨ï

ä®à¬ã«¨àã¥âáï â ª. ‘«¥¢

¨¬¥¥âáï ¯ ¤ îé ï ¨ ®âà ¦¥-- ï ¢®«- ,

á¯à ¢ | ¯à®è¥¤è ï. Аᨬ¯â®â¨ª¨ ¢®«-®¢®© äã-ªæ¨¨ â ª®¢ë:

8

eikx + A e

ikx;

= p

ei

!t

2

mE

¯à¨

x

!

eik1x−

hk

! −1

B

hk1 = 2m(E

V ) ¯à¨ x

:

<

;

−

!

+

1

:

q

•«®â-®á⨠x-ª®¬¯®-¥-â ⮪ à ¢-ë:

j¯ ¤ = hkm ; j®âà = −jAj2 hkm ; j¯à®è = jBj2 hkm1 :

Š®íää¨æ¨¥-âë ¯à®å®¦¤¥-¨ï D ¨ ®âà ¦¥-¨ï R à ¢-ë:

D = jj¯à®è¯ ¤ = kk1 jBj2; R = jjj¯®âà¤j = jAj2; R + D = 1 :

24

U (x) =

Ž¯â¨ç¥áª¨© - «®£ | ®âà ¦¥-¨¥ ᢥ⠯ਠ-®à¬ «ì-®¬ ¯ ¤¥-¨¨ - ¯«®áªãî £à -¨æã à §¤¥« ¤¢ãå á।. ‚ ®¯â¨ª¥ ¢®«-®¢®© ¢¥ªâ®à

k 2 = !c n;

£¤¥ n | ¯®ª § â¥«ì ¯à¥«®¬«¥-¨ï. ‡¤¥áì - 襩 § ¤ ç¥ á®®â¢¥âáâ¢ã¥â

á¨âã æ¨ï, ª®£¤ á¯à ¢	| ¢ ªãã¬,	á«¥¢ | á⥪«®.
‚ á«ãç ¥ 0 < E < V	ᨬ¯â®â¨ª	¯à¨ x ! +1 ¨§¬¥-ï¥âáï,

q

! ei!t B e− x; h = 2m(V − E) ;

‡¤¥áì ®¯â¨ç¥áª¨© - «®£ | ¯®«-®¥ ¢-ãâà¥--¥¥ ®âà ¦¥-¨¥.

‚Ž••Ž‘›

9.1. — áâ¨æ - 室¨âáï ¢ ¯®«¥ U(x) = −G (x). •à¨ t = 0 ¢®«-®¢ ï
äã-ªæ¨ï ¨¬¥¥â ¢¨¤	(	0) = e x =b	p
	x;	−j j	= b. • ©â¨ ¢¥à®ïâ-®áâì ⮣®, çâ®
¯à¨ t ! 1 ç áâ¨æ	®ª ¦¥âáï ¢ ®á-®¢-®¬ á®áâ®ï-¨¨ 0(x).
9.2. ’®â ¦¥ ¢®¯à®á ¤«ï £ ମ-¨ç¥áª®£® ®á樫«ïâ®à ¯à¨
			e−x2=(2b2)
		(x; 0) =				:
			( b2)1=4

9.3.• ©â¨ äã-ªæ¨î ƒà¨- ¤«ï ᢮¡®¤-®© ç áâ¨æë.

9.4.• ©â¨ D ¨ R ¤«ï ç áâ¨æë ¢ ¯®«¥ à¨á. 3

8

< 0 ¯à¨ x < 0

: V ¯à¨ x > 0 :

“ª § âì ®¯â¨ç¥áªãî - «®£¨î. ˆ§¢¥áâ-®, çâ® ¯à¨ ®âà ¦¥-¨¨ ®â ®¯â¨ç¥áª¨ ¡®«¥¥ ¯«®â-®© áà¥¤ë ¯à®¨á室¨â ¯®â¥àï ¯®«ã¢®«-ë. —¥- ¬ã ᮮ⢥âáâ¢ã¥â í⮠¥-¨¥ ¢ ¤ --®© § ¤ ç¥? • áᬮâà¥âì ¯à¥¤¥« h ! 0.

9.5. • ©â¨ D ¤«ï ç áâ¨æë ¢ ¯®«¥ ¯àאַ㣮«ì-®© ¯®â¥-æ¨ «ì-®© ï¬ë £«ã¡¨-ë V ¨ è¨à¨-ë a (à¨á. 4). „ âì £à 䨪 D(E), 㪠§ âì ãá«®¢¨¥ ¯à®§à ç-®áâ¨. “ª § âì -¥®¡å®¤¨¬®¥ ãá«®¢¨¥ ¯à®§à ç-®á⨠¢ á«ãç ¥ ¯®«ï à¨á. 5:

U(x) =	8	0		¯à¨ x < 0
	>		V1	¯à¨ 0 < x < a
	>	−
	>
	<	−V2
	>			¯à¨ x > a ;
	>

>

:

25

•¨á. 3: •®â¥-æ¨ «ì- ï \áâ㯥-ìª "

•¨á. 4: •à®å®¦¤¥-¨¥ ç áâ¨æë - ¤ ®¤-®¬¥à-®© ¯àאַ㣮«ì-®© אַ©

ᮮ⢥âáâ¢ãî饣® ¯à¨ V1 < V2 ¯à®á¢¥â«¥--®© ®¯â¨ª¥.

9.6.• ©â¨ D(E) ¤«ï ç áâ¨æë ¢ ¯®«¥ ¯àאַ㣮«ì-®£® ¯®â¥-æ¨ «ì-

-®£® ¡ àì¥à ¢ëá®â®î V ¨ è¨à¨-®î a (à¨á. 6), ®á®¡® à áᬮâà¥âì á«ãç © E < V; a 1.

9.7. • áᬮâà¥âì à áá¥ï-¨¥ ¢ ¯®«¥ U (x) = −G (x). Ž¡à â¨âì ¢-¨- ¬ -¨¥ - ¯®¢¥¤¥-¨¥ ¬¯«¨â㤠®âà ¦¥--®© ¨ ¯à®è¥¤è¥© ¢®«- ¯à¨ ¯à®¤®«¦¥-¨¨ à¥è¥-¨ï ¢ ®¡« áâì E < 0.

•¨á. 5: •®â¥-æ¨ «ì- ï í-¥à£¨ï, ᮮ⢥âáâ¢ãîé ï á«ãç î ¯à®á¢¥â«¥--®© ®¯â¨- ª¨

26

•¨á. 6: ’ã--¥«¨à®¢ -¨¥ ç áâ¨æë ç¥à¥§ ®¤-®¬¥à-ë© ¯àאַ㣮«ì-ë© ¡ àì¥à

x10. Š®¬¬ãâ â®àë.

‘-®¢ á®®â-®è¥-¨¥ -¥®¯à¥¤¥«¥--®á⥩. “à ¢-¥-¨¥ •à¥-ä¥áâ . ’¥®à¥¬ ® ¢¨à¨ «¥

ˆ§¬¥à¨¬®áâì ¢¥«¨ç¨-

…᫨ ¢¥«¨ç¨-ë A ¨ B ®¤-®¢à¥¬¥--® ¨§¬¥à¨¬ë, â® áãé¥áâ¢ã¥â ¯®«- - ï á¨á⥬ ¢®«-®¢ëå äã-ªæ¨© n, â ª¨å, çâ® n | ®¤-®¢à¥¬¥--® ᮡá⢥-- ï äã-ªæ¨ï ¨ A^, ¨ B^. •® ⮣¤

X	X	X
A^B^ = A^B^ cn n =		cnab n = cnB^A^ n = B^A^ ;
n	n	n
â® ¥áâì	[A;^ B^] A^B^ − B^A^ = 0:
ˆ ®¡à â-®, ¥á«¨ [A;^ B^] = 0, â® A^ ¨ B^ ¬®£ãâ ¨¬¥âì ®¡éãî á¨á⥬ã
ᮡá⢥--ëå äã-ªæ¨©. •ãáâì		a | ᮡá⢥-- ï äã-ªæ¨ï A^:
	A^	a = a a ;
⮣¤
	B^A^ a = aB^ a = A^B^ a ;

â® ¥áâì B^ a | ⮦¥ ᮡá⢥-- ï äã-ªæ¨ï A^ á ᮡá⢥--ë¬ §- ç¥- -¨¥¬ a. …᫨ ᯥªâà -¥¢ë஦¤¥-, ®âáî¤ á«¥¤ã¥â, çâ® B^ a á â®ç- -®áâìî ¤® ¬-®¦¨â¥«ï ᮢ¯ ¤ ¥â á a, â® ¥áâì B^ a = b a, â ª çâ® a, ¤¥©á⢨⥫ì-®, ï¥âáï ᮡá⢥--®© äã-ªæ¨¥© ®¯¥à â®à B^ (á ᮡá⢥--ë¬ §- ç¥-¨¥¬ b). ‚ á«ãç ¥ ¢ë஦¤¥--®£® ᯥªâà ¬®¦-® ¢ë¡à âì â ª¨¥ «¨-¥©-ë¥ ª®¬¡¨- 樨 Pi ci ai ᮡá⢥--ëå äã-ªæ¨© ®¯¥à â®à A^, ª®â®àë¥ ¡ã¤ãâ ®¤-®¢à¥¬¥--® ᮡá⢥--묨 äã-ªæ¨ï-

¬ B^.

• áᬮâà¨â¥ â ª¦¥ á«ãç © a = b = 0.

27

‘®®â-®è¥-¨¥ -¥®¯à¥¤¥«¥--®á⥩

Ž¯à¥¤¥«¨¬ ¤¨á¯¥àá¨î

r

A = h(A − hAi)2i :

•ãáâì íà¬¨â®¢ë ®¯¥à â®àë A^ ¨ B^ -¥ ª®¬¬ãâ¨àãîâ, [A;^ B^] = iC^ ¨ ¤«ï ¯à®áâ®âë hnjA^jni = hnjB^jni = 0. • áᬮâਬ á®áâ®ï-¨¥

jmi = A^ + iB^ jni :

Ÿá-®, çâ®

J( ) = hmjmi = hnj( A^ − iB^)( A^ + iB^)jni =

hnj 2A^2 + i (A^B^ − B^A^) + B^2jni = = 2 A2 − hCi + B2 0 :

Žâáî¤ á«¥¤ã¥â, çâ® A2 B2 1hCi2: ’ ª¨¬ ®¡à §®¬,

4

A B 1jhCij :

2

Š¢ -â®¢ë¥ áª®¡ª¨ •ã áá®-

Ž¯¥à â®à ¯à®¨§¢®¤-®© ¯® ¢à¥¬¥-¨ ddtA^ , ¯® ®¯à¥¤¥«¥-¨î, 㤮¢«¥â¢®- àï¥â ãá«®¢¨î

*

dA^

+

d

j

A^

i

:

dt

dth

j

ˆá¯®«ì§ãï “˜, ¯à ¢ãî ç

áâì

í⮣® à ¢¥-áâ¢

¬®¦-® ¯¥à¥¯¨á âì ¢

¢¨¤¥

d

A^

=

@

A^

+

@A^

+

A^

@

=

@A^

+

1

[A^H^ ]

:

dth j

* @t

@t

@t

+

ih

j i

+ *

+ *

*

@t

+

’ ª¨¬ ®¡à §®¬,

dA^

=

@A^

+

i

[H;^ A^] :

dt

@t

h

Š¢ -⮢ë©

- «®£ ᪮¡ª¨ •ã áá®-

fH; Ag ¢ëà ¦ ¥âáï ç¥à¥§ ª®¬-

¬ãâ â®à:

fH; Ag ! hi [H;^ A^] :

28

•®ª ¦¨â¥, çâ®
	d		@U
		p^x = −		:
	dt		@x
Žâáî¤ á«¥¤ã¥â ⥮६ •à¥-ä¥áâ

mdtd22 hri = −hrUi :

Ž¡á㤨⥠¥e á®®®â-®è¥-¨¥ á® ¢â®àë¬ § ª®-®¬ •ìîâ®- .

’¥®à¥¬ ® ¢¨à¨ «¥

•à¥¤¢ à¨â¥«ì-ë¥ ¯®«¥§-ë¥ á®®â-®è¥-¨ï:

[A;^ B^C^] = B^[A;^ C^] + [A;^ B^]C^; [H;^ p]^ = ihrU; [H;^ r] = −

ih

p^ :

m

•ãáâì jni | áâ 樮- à-®¥ á®áâ®ï-¨¥ ¤¨áªà¥â-®£® ᯥªâà

(ä¨-

-¨â-®¥ ¤¢¨¦¥-¨¥), ⮣¤

hnj[H;^ A^]jni = hnjH^A^jni − hnjA^H^ jni = (En − En)hnjA^jni = 0 :

‚ ç áâ-®áâ¨,

0 = n

[H;^ pr]^

n

=

n [H;^ p]r^

+ p[^H;^ r] n

= ih *n

r

r

U

−

p^2

n+

:

h j

j i

h j

j i

m

’ ª¨¬ ®®¡à §®¬,

2 n

p^2

n = n r U n :

*

2

m

+

h j r

j i

‚¥«¨ç¨- rrU - §ë¢ ¥âáï ¢¨à¨ «®¬ ¤ --®© ¬¥å -¨ç¥áª®© á¨á⥬ë. …᫨ ¯®â¥-æ¨ «ì- ï í-¥à£¨ï | ®¤-®à®¤- ï äã-ªæ¨ï ª®®à¤¨- â, â® ¥áâì ¥á«¨ U( r) = k U(r), â®, ¯® ⥮६¥ •©«¥à ®¡ ®¤-®à®¤-ëå

äã-ªæ¨ïå, rrU = k U , ¨«¨

2 n

p^2

n = k

h

n U

n

i

; n

p^2

n

=

k

En ; n U

n

i

=

2

En :

2m

2m

k + 2

k + 2

*

+

j j

*

+

h j j

•à¨¬¥àë

„«ï £ ମ-¨ç¥áª®£® ®á樫«ïâ®à k = 2, ¯®í⮬ã

n m!2x2 n		=	n	p^2		n = h!(n + 1):
D		E	*	m			+	2

29

„«ï ⮬ ¢®¤®à®¤ k = −1, ¯®í⮬ã

	*n		e2	n+
			r

=	*n	p^2	n+	=	−	2En :	(10:1)
		m

‚Ž••Ž‘›

10.1.Ž¡êïá-¨â¥, ¯®ç¥¬ã ⥮६ ® ¢¨à¨ «¥ -¥ ¨¬¥¥â ¬¥áâ ¤«ï ¨-ä¨-¨â-®£® ¤¢¨¦¥-¨ï.

10.2.• ©â¨ á®®â-®è¥-¨¥ -¥®¯à¥¤¥«¥--®á⥩ ¤«ï x ¨ K, ¤«ï

U ¨ K, £¤¥ K = p2=(2m).

10.3. „«ï ç áâ¨æë, - 室ï饩áï ¢ á®áâ®ï-¨¨ (x; y; z), - ©â¨ ¢¥- à®ïâ-®áâì ⮣®, çâ® ¥¥ ª®®à¤¨- â x ¨ ¨¬¯ã«ìá py à ᯮ«®¦¥-ë ¢

¯à¥¤¥« å x1 < x < x2; py1 < py < py2.
10.4. „«ï £ ¬¨«ìâ®-¨ - H^	= p^2=(2m) + U (r) - ©â¨ ª®¬¬ãâ â®à
[H;^ r]. ˆá¯®«ì§ãï íâ®â १ã«ìâ	â, ¯®ª § âì, çâ® á।-¥¥ §- ç¥-¨¥ ¨¬-

¯ã«ìá ç áâ¨æë ¤«ï áâ 樮- à-®£® á®áâ®ï-¨ï ¢ á«ãç ¥ ä¨-¨â-®£® ¤¢¨¦¥-¨ï à ¢-® -ã«î h E j p^ j E i = 0.

x11. Ž¯¥à â®à ᤢ¨£ . ’¥®à¥¬				•«®å
Ž¯¥à â®à ᤢ¨£ T^a ®¯à¥¤¥«ï¥âáï á®®â-®è¥-¨¥¬
T^a (x) (x + a) :
’ ª ª ª		an			!n (x) ;
(x + a) = 1				d
		!
n	=0 n			dx
X

â® ®¯¥à â®à ᤢ¨£ ¬®¦¥â ¡ëâì ¢ëà ¦¥- ç¥à¥§ ®¯¥à â®à ¨¬¯ã«ìá

T^a = eiap=^ h :

Ž¡à ⨬ ¢-¨¬ -¨¥ - â®, çâ® ¯à¨ ¡¥áª®-¥ç-® ¬ «®¬ ᤢ¨£¥ a ! 0 ®¯¥à â®à ᤢ¨£ ¨¬¥¥â ¢¨¤

a T^ a = 1 + i h p^;

â® ¥áâì ®¯¥à â®à ¨¬¯ã«ìá p^x ï¥âáï ¨-ä¨-¨â¥§¨¬ «ì-ë¬ ®¯¥à - â®à®¬ ¤«ï ᤢ¨£ ¢¤®«ì ®á¨ x.

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