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Файл:4 семестр (2 курс)лала / Физика / Кратко / 17 Собственные значения ... модуля
.txt яю1 7 . !>1AB25==K5 7=0G5=8O 8 A>1AB25==K5 DC=:F88 <>4C;O <><5=B0 8<?C;LAO 8 53> ?@>5:F88 =0 :>>@48=0B=CN >AL z .
><5=B 8<?C;LA0 G0AB8FK L >B=>A8B5;L=> =0G0;0 :>>@48=0B 2 :;0AA8G5A:>9 <5E0=8:5 >?@545;O5BAO 25:B>@=K< ?@>872545=85< [ ( 25:B>@ r ) , ( 25:B>@ p ) ] , B. 5.
| ( 25:B>@ i ) ( 25:B>@ j ) ( 25:B>@ k ) |
( 25:B>@ L ) = | x y z |
| p ( x - 2>5) p ( y - 2>5) p ( s - =>5) |
"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 r 8 p 8<5;8 >?@545;5==K5 7=0G5=8O.
0AA<>B@8< <><5=B 8<?C;LA0 :20=B>2>9 G0AB8FK. 0A:@K20O 25:B>@=>5 ?@>872545=85 [ ( 25:B>@ r ) , ( 25:B>@ p ) ] , =0E>4OB >?5@0B>@K ?@>5:F89 <><5=B0 8<?C;LA0 =0
:>>@48=0B=K5 >A8 X , Y , Z . 0?@8<5@, =0 >AL Z :
( 25:B>@ L A :@KH5G:>9 A25@EC) = i * h ( A G5@B>9) * ( y * d / d x - x * d / d y ) / / d - =5?>;=0O ?@>872>4=0O.
'5@57 MB8 ?@>5:F88 >?5@0B>@ 25:B>@0 <><5=B0 8<?C;LA0 2K@0605BAO :0::
( 25:B>@ L A :@KH5G:>9 A25@EC) = ( L A :@KH5G:>9 A25@EC x - 2>5) * ( 25:B>@ i ) + ( L A :@KH5G:>9 A25@EC C- 2>5) * ( 25:B>@ j ) + ( L A :@KH5G:>9 A25@EC s - =>5) * ( 25:B>@ k ) .
40;L=59H5< 1C45< 8A?>;L7>20BL >?5@0B>@ ?@>5:F88 <><5=B0 8<?C;LA0 =0 >AL Z , => =5 2 45:0@B>2>9, 0 2 AD5@8G5A:>9 A8AB5<5 :>>@48=0B ( r , " > A ?5@5:;048=:>9" , " D8" ) :
( L A :@KH5G:>9 A25@EC s - =>5) = - i * h ( A G5@B>9) * d / d " D8" / / d - >?OBL =5?>;=0O ?@>872>4=0O.
?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. !>1AB25==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.
>6=> ?@>25@8BL 8 C1548BLAO, GB> >?5@0B>@K ?@>5:F89 <><5=B0 8<?C;LA0 =5 :><<CB8@CNB <564C A>1>9, ?>MB><C =5 ACI5AB2C5B A>AB>O=8O, 2 :>B>@>< 1K 2A5 B@8 8 4065 :0:85- ;81> 425 87 B@5E ?@>5:F89 L x , L y , L z 8<5;8 >?@545;5==K5 7=0G5=8O, >B;8G=K5 >B =C;O.
B<5B8<, GB> 2 >B;8G85 >B <><5=B0 8<?C;LA0 C 8<?C;LA0 >4=>2@5<5==> 87<5@8<K B@8 :><?>=5=BK p x , p y , p z .
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 ( 25:B>@ L ) = 0 8 2A5 B@8 ?@>5:F88 >4=>2@5<5==> @02=K =C;N:
L x = L y = L z = 0 .
>4C;L <><5=B0 8<?C;LA0 4286CI59AO <8:@>G0AB8FK:
L = h ( A G5@B>9) * s q r t ( l * ( l + 1 ) ) , l = 0 , 1 , 2 . . 84=>, GB> MB0 25;8G8=0 O2;O5BAO 48A:@5B=>9 ( :20=B>20==>9) .
@>5:F8O <><5=B0 8<?C;LA0 L z .
L z = m * h ( A G5@B>9) , m = 0 , + - 1 , + - 2 , . . , 345 G8A;> m =07K20NB <03=8B=K< :20=B>2K< G8A;><. "0:8< >1@07><, ?>AB>O==CN ;0=:0 h ( A G5@B>9) <>6=> @0AA<0B@820BL :0: 5AB5AB25==CN 548=8FC C3;>2>3> <><5=B0.
B<5B8<, GB> MB> 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 .
=> >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=BC5BAO.
><5=B 8<?C;LA0 G0AB8FK L >B=>A8B5;L=> =0G0;0 :>>@48=0B 2 :;0AA8G5A:>9 <5E0=8:5 >?@545;O5BAO 25:B>@=K< ?@>872545=85< [ ( 25:B>@ r ) , ( 25:B>@ p ) ] , B. 5.
| ( 25:B>@ i ) ( 25:B>@ j ) ( 25:B>@ k ) |
( 25:B>@ L ) = | x y z |
| p ( x - 2>5) p ( y - 2>5) p ( s - =>5) |
"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 r 8 p 8<5;8 >?@545;5==K5 7=0G5=8O.
0AA<>B@8< <><5=B 8<?C;LA0 :20=B>2>9 G0AB8FK. 0A:@K20O 25:B>@=>5 ?@>872545=85 [ ( 25:B>@ r ) , ( 25:B>@ p ) ] , =0E>4OB >?5@0B>@K ?@>5:F89 <><5=B0 8<?C;LA0 =0
:>>@48=0B=K5 >A8 X , Y , Z . 0?@8<5@, =0 >AL Z :
( 25:B>@ L A :@KH5G:>9 A25@EC) = i * h ( A G5@B>9) * ( y * d / d x - x * d / d y ) / / d - =5?>;=0O ?@>872>4=0O.
'5@57 MB8 ?@>5:F88 >?5@0B>@ 25:B>@0 <><5=B0 8<?C;LA0 2K@0605BAO :0::
( 25:B>@ L A :@KH5G:>9 A25@EC) = ( L A :@KH5G:>9 A25@EC x - 2>5) * ( 25:B>@ i ) + ( L A :@KH5G:>9 A25@EC C- 2>5) * ( 25:B>@ j ) + ( L A :@KH5G:>9 A25@EC s - =>5) * ( 25:B>@ k ) .
40;L=59H5< 1C45< 8A?>;L7>20BL >?5@0B>@ ?@>5:F88 <><5=B0 8<?C;LA0 =0 >AL Z , => =5 2 45:0@B>2>9, 0 2 AD5@8G5A:>9 A8AB5<5 :>>@48=0B ( r , " > A ?5@5:;048=:>9" , " D8" ) :
( L A :@KH5G:>9 A25@EC s - =>5) = - i * h ( A G5@B>9) * d / d " D8" / / d - >?OBL =5?>;=0O ?@>872>4=0O.
?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. !>1AB25==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.
>6=> ?@>25@8BL 8 C1548BLAO, GB> >?5@0B>@K ?@>5:F89 <><5=B0 8<?C;LA0 =5 :><<CB8@CNB <564C A>1>9, ?>MB><C =5 ACI5AB2C5B A>AB>O=8O, 2 :>B>@>< 1K 2A5 B@8 8 4065 :0:85- ;81> 425 87 B@5E ?@>5:F89 L x , L y , L z 8<5;8 >?@545;5==K5 7=0G5=8O, >B;8G=K5 >B =C;O.
B<5B8<, GB> 2 >B;8G85 >B <><5=B0 8<?C;LA0 C 8<?C;LA0 >4=>2@5<5==> 87<5@8<K B@8 :><?>=5=BK p x , p y , p z .
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 ( 25:B>@ L ) = 0 8 2A5 B@8 ?@>5:F88 >4=>2@5<5==> @02=K =C;N:
L x = L y = L z = 0 .
>4C;L <><5=B0 8<?C;LA0 4286CI59AO <8:@>G0AB8FK:
L = h ( A G5@B>9) * s q r t ( l * ( l + 1 ) ) , l = 0 , 1 , 2 . . 84=>, GB> MB0 25;8G8=0 O2;O5BAO 48A:@5B=>9 ( :20=B>20==>9) .
@>5:F8O <><5=B0 8<?C;LA0 L z .
L z = m * h ( A G5@B>9) , m = 0 , + - 1 , + - 2 , . . , 345 G8A;> m =07K20NB <03=8B=K< :20=B>2K< G8A;><. "0:8< >1@07><, ?>AB>O==CN ;0=:0 h ( A G5@B>9) <>6=> @0AA<0B@820BL :0: 5AB5AB25==CN 548=8FC C3;>2>3> <><5=B0.
B<5B8<, GB> MB> 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 .
=> >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=BC5BAO.
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