4 семестр (2 курс)лала / Физика / Кратко / 16 Операторы физических величин

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"0:>5 >?@545;5=85 2 :20=B>2>9 <5E0=8:5 =5 8<55B A<KA;0, B0: :0: =5 ACI5AB2C5B A>AB>O=8O, 2 :>B>@>< 1K >10 25:B>@0 rr 8 p r 8<5;8 >?@545;5==K5 7=0G5=8O.

0AA<>B@8< <><5=B 8<?C;LA0 :20=B>2>9 G0AB8FK.  :20=B>2>9 <5E0=8:5 25:B>@=><C ?@>872545=8N [rr, pr] A>>B25BAB2C5B >?5@0B>@ [rr"?A8", pr"?A8"].



?5@0B>@ <><5=B0 8<?C;LA0 7028A8B B>;L:> >B =0?@02;5=8O :>>@48=0B=KE >A59. >MB><C 53> B0:65 =07K20NB >?5@0B>@>< C3;>2>3> <><5=B0. !>1AB-

25==K5 7=0G5=8O >?5@0B>@>2 ?@>5:F89 <><5=B0 8<?C;LA0 B>65 =5 7028AOB >B 2K1>@0 =0G0;0 :>>@48=0B.



B0:, =5 ACI5AB2C5B B0:>3> A>AB>O=8O :20=B>2>9 G0AB8FK, 2 :>B>@>< 1K 25:B>@ <><5=B0 8<?C;LA0 8<5; >?@545;5==>5 7=0G5=85, B. 5. 1K; 1K ?>;-

=>ABLN >?@545;5= :0: ?> 25;8G8=5, B0: 8 ?> =0?@02;5=8N. A:;NG5=85< O2;O5BAO B>;L:> A;CG09, :>340 0 = L

8 2A5 B@8 ?@>5:F88 >4=>2@5<5==> @02=K =C;N:

Lx = Ly = Lz = 0 .

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>6=> ?>:070BL, GB> 4;O A>1AB25==KE 7=0G5=89 >?5@0B>@0 "?A8"L A?@0254;82>

L2 = l(l +1)h2 ,

345 l  >@18B0;L=>5 (078<CB0;L=>5) :20=B>2>5 G8A;>. BAN40 <>4C;L <><5=B0 8<?C;LA0 4286CI59AO <8:@>G0AB8FK

L = h l(l +1) , l = 0,1, 2, .... (27.110)

84=>, GB> MB0 25;8G8=0 O2;O5BAO 48A:@5B=>9 (:20=B>20==>9).

B<5B8<, GB> >?5@0B>@K "?A8"

Lx , "?A8"

Ly 8 "?A8"

Lz :><<CB8@CNB A "?A8"L . !;54>20-

B5;L=>, <>6=> >4=>2@5<5==> >?@545;8BL 25;8G8=C <><5=B0 8<?C;LA0 L (8;8 55 :204@0B L2 ) 8 >4=C 87 53> ?@>5:F89 ( Lx , Ly 8;8 Lz ). 1KG=> @0AA<0B@820NB

?@>5:F8N =0 >AL Z, B0: :0: 2 MB>< A;CG05 >?5@0B>@ "?A8" z L 70405BAO 1>;55 ?@>AB>9

D>@<C;>9.

@>5:F8O <><5=B0 8<?C;LA0 Lz . 'B>1K >?@545;8BL A>1AB25==K5 7=0G5=8O 8 A>1AB25==K5 DC=:F88 >?5@0B>@0 C3;>2>3> <><5=B0 G0AB8FK, =5>1E>48<>,

A>3;0A=> 2K@065=8N, @5H8BL C@02=5=85 "?A8" -i h(A G5@B>9)d/dD8 * ?A8 = Lz ?A8

345 2>;=>20O DC=:F8O O2;O5BAO DC=:F859 AD5@8G5A:8E :>>@48=0B:

y =y (r,q,j) . >4AB0=>2:0 y = Ceaj (C = C(r,q ) ) ?@82>48B ?>A;5 A>:@0I5=8O

=0 >1I89 <=>68B5;L Ceaj : C@02=5=8N

z - iha = L , z ha = iL .



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=>20O DC=:F8O =5 4>;6=0 87<5=OBLAO: y (j + 2p ) =y (j) . >A:>;L:C DC=:F8O

eia ?5@8>48G=0 A ?5@8>4>< 2p , B> A>3;0A=> MB> @025=AB2> <>65B 2K-

?>;=OBLAO B>;L:> ?@8 CA;>288

i Lz 2p = m"?A8" 2pi

.

>MB><C

Lz = mh , m = 0, ± 1, ± 2

345 G8A;> B =07K20NB <03=8B=K< :20=B>2K< G8A;><. "0:8< >1@07><, ?>AB>-

O==CN ;0=:0 h <>6=> @0AA<0B@820BL :0: 5AB5AB25==CN 548=8FC C3;>2>3>

<><5=B0. B<5B8<, GB> C@02=5=85 70405B A?5:B@ @07@5H5==KE 7=0G5=89

?@>5:F88 <><5=B0 8<?C;LA0 =0 2K45;5==CN >AL Z.

025=AB2> >7=0G05B, GB>, ?>A:>;L:C =0?@02;5=85 >A8 Z 2K18@0NB

?@>872>;L=>, B> ?@>5:F8O C3;>2>3> <><5=B0 =0 ;N1>5 =0?@02;5=85 :20=BC-

5BAO, @8A. 27.1. 07C<55BAO, AE5<0B8G5A:>5 87>1@065=85 =0 @8A. 27.1 =5 A;54C5B

?>=8<0BL 1C:20;L=>, B0: :0: «25:B>@» L

?@8=F8?80;L=> =5 8<55B >?@545;5=-

=KE =0?@02;5=89 2 ?@>AB@0=AB25. B<5B8<, GB> ?@8 >?@545;5==>< 7=0G5=88

<>4C;O C3;>2>3> <><5=B0 8 >?@545;5==>< 7=0G5=88 ?@>5:F88 Lz ?@>5:F88 Lx

8 Ly =5 8<5NB >?@545;5==KE 7=0G5=89 (70 8A:;NG5=85< A;CG0O, :>340 2A5 B@8

:><?>=5=BK C3;>2>3> <><5=B0 >4=>2@5<5==> @02=K =C;N). =0G5=8O L 8 Lz, >B-

;8G=K5 >B (27.110) 8 (27.13), =5 <>3CB =01;N40BLAO =8 ?@8 :0:8E CA;>28OE.

@>5:F8O ;N1>3> 25:B>@0 =5 <>65B 1KBL 1>;LH5 <>4C;O MB>3> 25:B>@0,

B. 5. Lz "?A8" L. >MB><C 2 A>>B25BAB288 A D>@<C;0<8 (27.110) 8 (27.13) 2K?>;=O-

5BAO CA;>285

m < l(l +1) ; (27.14)

A;54>20B5;L=>, <0:A8<0;L=>5 7=0G5=85 m @02-

=> l 8 <>6=> 70?8A0BL, GB>

m = 0, ±1, ± 2, ..., ± l .

@8 7040==>< l G8A;> B ?@8=8<05B (2l +1) 7=0-

G5=89:

- l, - (l -1), ..., 0, ..., l -1, l ,

>1@07CNI8E A?5:B@ ?@>5:F88 L mh z = =0 ;N-

1CN 2K45;5==CN >AL Z,



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2A5 2>7<>6=K5 7=0G5=8O 53> ?@>5:F88 =0 >AL Z. "0:, =0?@8<5@, :>340 >@18B0;L-

=>5 :20=B>2>5 G8A;> l = 2, B>

L = h 6 , Lz = -2h, -1h, 0,1h, 2h .

>;CG5==K5 @57C;LB0BK, >?@545;ONI85 2>7<>6=K5 7=0G5=8O L 8 Lz, =07K20NB

?@>AB@0=AB25==K< :20=B>20=85<. ;O =03;O4=>AB8 ?@>AB@0=AB25==>5 :20=-

B>20=85 >1KG=> ?@54AB02;ONB 3@0D8G5A:8, :0: =0 @8A. 27.1: ?> >A8 Z >B:;04K-

20NB 2>7<>6=K5 7=0G5=8O mh , @0AA<0B@820O 8E :0: ?@>5:F88 =0 >AL Z 25:B>@0

L r

4;8=K h l(l +1) .