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Файл:4 семестр (2 курс)лала / Физика / Кратко / 27 Квантово-механическая моде
.txt яю2 7 . 20=B>2>- <5E0=8G5A:0O <>45;L 0B><0 2>4>@>40 ( @57C;LB0BK @5H5=8O C@02=5=8O (@Q48=35@0 4;O F5=B@0;L=>3> A8;>2>3> ?>;O) . 20=B>2K5 G8A;0 M;5:B@>=0 2 0B><5.
C45< @0AA<0B@820BL 2>4>@>4>?>4>1=K9 0B><, A>45@60I89 548=AB25==K9 2=5H=89 M;5:B@>=.
-;5:B@8G5A:>5 ?>;5, A>740205<>5 O4@><, O2;O5BAO ?@8<5@>< F5=B@0;L=>3> ?>;O. >A:>;L:C <0AA0 O4@0 2>4>@>4=>?>4>1=>3> 0B><0 2> <=>3> @07 1>;LH5 <0AAK M;5:B@>=0 ( @07;8G85 4;O 0B><0 2>4>@>40 A>AB02;O5B ?>GB8 2 0 0 0 @07) , B> ?@81;865==> <>6=> AG8B0BL, GB> >=0 15A:>=5G=> 25;8:0 8 GB> O4@> 2A5 2@5<O =0E>48BAO 2 =0G0;5 :>>@48=0B ( O4@> =5?>4286=>) .
">340 7040GC > 2>4>@>4>?>4>1=>< 0B><5 <>6=> ?@54AB028BL :0: 7040GC > 42865=88 M;5:B@>=0 2 D8:A8@>20==>< ?>;5 A8; A ?>B5=F80;><:
U ( r ) = - k * ( Z e ^ 2 ) / r = - ( Z e ^ 2 ) / 4 * ?8* ( M?A8;>= =C;52>5) * r , 345 r - @0480;L=0O ?5@5<5==0O - @0AAB>O=85 <564C M;5:B@>=>< 8 O4@><, Z - 0B><=>5 G8A;>, e - 01A>;NB=>5 7=0G5=85 70@O40 M;5:B@>=0, ?>B5=F80; U ( r ) >?8AK205B :C;>=>2A:>5 ?>;5 O4@0.
2865=85 M;5:B@>=0 2 B0:>< ?>;5 <>6=> @0AA<0B@820BL :0: 42865=85 2 =5:>B>@>9 AD5@8G5A:>9 ?>B5=F80;L=>9 O<5 ( >1>945BAO 157 @8AC=:0) .
MB>< A;CG05 AB0F8>=0@=>5 C@02=5=85 (@548=35@0 8<55B 284:
" B@5C3>;L=8:" " ?A8" ( B0: =07K205<K9 B@57C15F) + 2 m ( e - B>5) / ( h A G5@B>9 2 :204@0B5) * ( E - U ) * " ?A8" = 0 , 345 2>;=>20O DC=:F8O " ?A8" ( x , y , z ) >?8AK205B A>AB>O=85 M;5:B@>=0 2 0B><5, E - 7=0G5=8O ?>;=>9 M=5@388 M;5:B@>=0 2 0B><5, :>B>@K5 B@51C5BAO =09B8 ?@8 CA;>288, GB> " ?A8" C4>2;5B2>@O5B B@51>20=8O< :>=5G=>AB8, >4=>7=0G=>AB8 8 =5?@5@K2=>AB8 ( M2> :0: x D ) .
>6=> ?>:070BL, GB> C@02=5=85 ( 2 8 . 1 1 0) 8<55B >4=>7=0G=K5, :>=5G=K5 8 =5?@5@K2=K5 @5H5=8O ?@8 ;N1KE ?>;>68B5;L=KE 7=0G5=8OE M=5@388, 0 B0:65 ?@8 48A:@5B=KE >B@8F0B5;L=KE 7=0G5=8OE M=5@388, @02=KE:
E ( n - =>5) = - ( k ^ 2 ) * m ( e - B>5) * ( e ^ 4 ) * ( Z ^ 2 ) / 2 * ( h A G5@B>9 2 :204@0B5) * ( n ^ 2 ) , n = 1 , 2 , 3 . .
20=B>2K5 G8A;0 0B><0 2>4>@>40.
!B0F8>=0@=K5 A>AB>O=8O 0B><0 2>4>@>40 157 CG5B0 A?8=0 <>6=> >E0@0:B5@87>20BL =01>@>< B@5E :20=B>2KE G8A5;.
20=B>2>5 G8A;> n 87 B5>@88 >@0 A>E@0=O5BAO 8 2 :20=B>2>9 <5E0=8:5 8 =07K205BAO + " + '! . => >?@545;O5B M=5@35B8G5A:85 C@>2=8 0B><0 2>4>@>40:
n = 1 , 2 , 3 , . . , 15A:>=5G=>ABL.
;O 0B><0 2>4>@>40 2 A>AB>O=88 A 7040==K< 3;02=K< :20=B>2K< G8A;>< n ", " '! l ?@8=8<05B 7=0G5=8O:
l = 0 , 1 , 2 , . . , ( n - 1 ) , >?@545;OO <>4C;L >@18B0;L=>3> <5E0=8G5A:>3> <><5=B0 0B><0, B. 5. <><5=B0 8<?C;LA0 M;5:B@>=0 >B=>A8B5;L=> O4@0:
M ( l - =>5) = h ( A G5@B>9) * ( :>@5=L 87 ( l * ( l + 1 ) ) ) .
03=8B=>5 :20=B>2>5 G8A;> m ( l - =>5) ?@8 40==>< l ?@8=8<05B 7=0G5=8O:
m ( l - =>5) = 0 , + - 1 , + - 2 , . . , + - l , 8 >?@545;O5B ?@>5:F8N >@18B0;L=>3> <5E0=8G5A:>3> <><5=B0 =0 ?@>872>;L=> 2K1@0==>5 =0?@02;5=85 Z :
M ( l z - B>5) = m ( l - =>5) * h ( A G5@B>9) .
C45< @0AA<0B@820BL 2>4>@>4>?>4>1=K9 0B><, A>45@60I89 548=AB25==K9 2=5H=89 M;5:B@>=.
-;5:B@8G5A:>5 ?>;5, A>740205<>5 O4@><, O2;O5BAO ?@8<5@>< F5=B@0;L=>3> ?>;O. >A:>;L:C <0AA0 O4@0 2>4>@>4=>?>4>1=>3> 0B><0 2> <=>3> @07 1>;LH5 <0AAK M;5:B@>=0 ( @07;8G85 4;O 0B><0 2>4>@>40 A>AB02;O5B ?>GB8 2 0 0 0 @07) , B> ?@81;865==> <>6=> AG8B0BL, GB> >=0 15A:>=5G=> 25;8:0 8 GB> O4@> 2A5 2@5<O =0E>48BAO 2 =0G0;5 :>>@48=0B ( O4@> =5?>4286=>) .
">340 7040GC > 2>4>@>4>?>4>1=>< 0B><5 <>6=> ?@54AB028BL :0: 7040GC > 42865=88 M;5:B@>=0 2 D8:A8@>20==>< ?>;5 A8; A ?>B5=F80;><:
U ( r ) = - k * ( Z e ^ 2 ) / r = - ( Z e ^ 2 ) / 4 * ?8* ( M?A8;>= =C;52>5) * r , 345 r - @0480;L=0O ?5@5<5==0O - @0AAB>O=85 <564C M;5:B@>=>< 8 O4@><, Z - 0B><=>5 G8A;>, e - 01A>;NB=>5 7=0G5=85 70@O40 M;5:B@>=0, ?>B5=F80; U ( r ) >?8AK205B :C;>=>2A:>5 ?>;5 O4@0.
2865=85 M;5:B@>=0 2 B0:>< ?>;5 <>6=> @0AA<0B@820BL :0: 42865=85 2 =5:>B>@>9 AD5@8G5A:>9 ?>B5=F80;L=>9 O<5 ( >1>945BAO 157 @8AC=:0) .
MB>< A;CG05 AB0F8>=0@=>5 C@02=5=85 (@548=35@0 8<55B 284:
" B@5C3>;L=8:" " ?A8" ( B0: =07K205<K9 B@57C15F) + 2 m ( e - B>5) / ( h A G5@B>9 2 :204@0B5) * ( E - U ) * " ?A8" = 0 , 345 2>;=>20O DC=:F8O " ?A8" ( x , y , z ) >?8AK205B A>AB>O=85 M;5:B@>=0 2 0B><5, E - 7=0G5=8O ?>;=>9 M=5@388 M;5:B@>=0 2 0B><5, :>B>@K5 B@51C5BAO =09B8 ?@8 CA;>288, GB> " ?A8" C4>2;5B2>@O5B B@51>20=8O< :>=5G=>AB8, >4=>7=0G=>AB8 8 =5?@5@K2=>AB8 ( M2> :0: x D ) .
>6=> ?>:070BL, GB> C@02=5=85 ( 2 8 . 1 1 0) 8<55B >4=>7=0G=K5, :>=5G=K5 8 =5?@5@K2=K5 @5H5=8O ?@8 ;N1KE ?>;>68B5;L=KE 7=0G5=8OE M=5@388, 0 B0:65 ?@8 48A:@5B=KE >B@8F0B5;L=KE 7=0G5=8OE M=5@388, @02=KE:
E ( n - =>5) = - ( k ^ 2 ) * m ( e - B>5) * ( e ^ 4 ) * ( Z ^ 2 ) / 2 * ( h A G5@B>9 2 :204@0B5) * ( n ^ 2 ) , n = 1 , 2 , 3 . .
20=B>2K5 G8A;0 0B><0 2>4>@>40.
!B0F8>=0@=K5 A>AB>O=8O 0B><0 2>4>@>40 157 CG5B0 A?8=0 <>6=> >E0@0:B5@87>20BL =01>@>< B@5E :20=B>2KE G8A5;.
20=B>2>5 G8A;> n 87 B5>@88 >@0 A>E@0=O5BAO 8 2 :20=B>2>9 <5E0=8:5 8 =07K205BAO + " + '! . => >?@545;O5B M=5@35B8G5A:85 C@>2=8 0B><0 2>4>@>40:
n = 1 , 2 , 3 , . . , 15A:>=5G=>ABL.
;O 0B><0 2>4>@>40 2 A>AB>O=88 A 7040==K< 3;02=K< :20=B>2K< G8A;>< n ", " '! l ?@8=8<05B 7=0G5=8O:
l = 0 , 1 , 2 , . . , ( n - 1 ) , >?@545;OO <>4C;L >@18B0;L=>3> <5E0=8G5A:>3> <><5=B0 0B><0, B. 5. <><5=B0 8<?C;LA0 M;5:B@>=0 >B=>A8B5;L=> O4@0:
M ( l - =>5) = h ( A G5@B>9) * ( :>@5=L 87 ( l * ( l + 1 ) ) ) .
03=8B=>5 :20=B>2>5 G8A;> m ( l - =>5) ?@8 40==>< l ?@8=8<05B 7=0G5=8O:
m ( l - =>5) = 0 , + - 1 , + - 2 , . . , + - l , 8 >?@545;O5B ?@>5:F8N >@18B0;L=>3> <5E0=8G5A:>3> <><5=B0 =0 ?@>872>;L=> 2K1@0==>5 =0?@02;5=85 Z :
M ( l z - B>5) = m ( l - =>5) * h ( A G5@B>9) .
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