perturbaciones estacionarias

! "#$%&'$&%! (")"%!* +&" ,-$&./0 ,# ") *! *"''/1) !)$"%/,% 2!%! "* "#3

2 "'$%, 4"* 5$,0, 4" 6 ',)',%4!-! -/") ',) *,# 4!$,# "72 "%/0 ")$!*"# +&" #"

8!-9!) ,-#"%.!4, ") "* 2%/0 "% '&!%$, 4"* #/(*, :: ; ,%!%,) *,# /)#$%&0 ")$,# ? $@')/'!# "72 "%/0 ")$!*"# #" ,-#"%.!%,) 4"3

$!**"# +&" ), 2,49!) "72*/'!%#" ',) *! $",%9! 4"#!%%,**!4! = ! "#! "#$%&'$&%!

05# 4"$!**!4! #" *" 4"),0 /)1 "#$%&'$&%! A)! 4"* 5$,0, 4" 6 ; B) "#$! *"''/1)

#" 2%"$")4" > $/A'!% $"1%/'!0 ")$" *! "#$%&'$&%! A)! /)$%,4&'/")4, 2"%$&%-!3

'/,)"# #,-%" *! "#$%&'$&%! ,-$")/4! ") *! *"''/1) !)$"%/,% ; C)$"# 4" "02"D!%

',) *,# $@%0 /),# +&" "72*/'!) *! "#$%&'$&%! A)! /)$%,4&'/%"0,# *! $",%9! 4"

2 "%$&%-!'/,)"# /)4"2 ")4/")$"# 4"* $/"02, +&" "# "* /)(%"4/")$" E&)4!0 ")$!*

") $,4, *, +&" #/(&" ;

F GH IJH I KILHIM IK IIIM IN

IOK

B) 0&'

8

!# #/$&!'/,)"# = ',0, *! +&" "#$!0,# "#$&4/!)4, = *! "#$%&'$&%!

-5#/'! 4"* 2%,-*"0 ! *! "72*/'! &)

8

!0 /*$,)/!), 4!4, = '&?,# !&$,.!*,%"# ?

!&$,."'$,%"# #" #!- ") ,-$")"% "7!'$!0 ")$" ;

H0(0)n = E

(0)n

(0)n .

,# 4" "**!# ; ,% "**, =

2%,2,)"0,# +&" 2,4"0,# ,-$")"% *!# #,*&'/,)"# 4"

',0, &) 4"#!%%,**,

") #"%/" 4" 2,$")'/!# ") "* 2!%50 "$%,

En = E(0)n + E

(1)n +

2E(2)n + . . .

n = (0)n +

(1)n +

2(2)n + . . .

B* ',)> &)$, 4" "#$!4,# |n(0) #,) &)! -!#" 2!%! "* "#2!'/, 4" 6/*- "%$ 4"*2%,-*"0 ! ;

H0?! "# 4/!(,)!* ") "#! -!#" = 2!%! ")',)$%!% *! #,*&'/1) "7!'$!

-!#$! 4/!(,)!*/D!% *! 2 "%$&%-!'/1)

W") *! -!#" |n(0) ; ! !2%,7/0 !'/1)

") 2%/0 "% ,%4") 4" $",%9! 4" 2 "%$&%-!'/,)"# ',)#/#$" ") 4"#2%"'/!% *,#

"*"0 ")$,# 4" 0 !$%/D ), 4/!(,)!*"# 4"

WE%")$" ! *,# 4/!(,)!*"# = 2 ,% *,

+&" *! #,*&'/1) !2%,7 /0 !4! ") "#$" ,%4") ! '!4!

E(0)n

"# (0)n|W |n(0) ;

1

E1(0) E2

(0)E3

(0)

2 3

2

3

1 2 W 2 3 W 2 2 W

1 2 W 1 3W

1 3 W 3 2 W 3 3 W

1 1 W

1

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

(0) (0) (0)

(0)

(0)

(0)

%

; "

"

! "'&!'/1) ") ,%4")

")

!)$"%/,% *! 2,4"0,# %""#'%/-/%

(H0 E(0)n

)(1)n =

(W E(1)n

)(0)n .

C4"0 5# = *!# E&)'/,)"#

(0)n

',)

n = 0, 1, 2, . . .E,%0 !) &)! -!#" 4"* "#2!'/,

E&)'/,)!* 4" )&"#$%, 2%,-*"0 ! ; ,% "**, = '&!*+&/"% E&)'/1) 4" "#$" "#2!'/,

#" 2&"4" "#'%/-/% ") $@%0 /),# 4" *!# E&)'/,)"# 4" *! -!#"

(1)n =p6=n

C(n)p (0)p .

#,) *, +&" *! "'&!'/1) !)$"%/,% +&"4!

(H0 E(0)n

)p6=n

C(n)p (0)p =

(W E(1)n

)(0)n .

&*$/2 */'!)4, "#$! "'&!'/1) 2,% *! /D+&/"%4! 2,%

(0)m

" /)$"(%!)4, !

$, 4, "* "#2!'/, ),# +&"4!

(E(0)m E(0)n

)C(n)m = (0)m |W |(0)n ,

4" 4,)4"

C(n)m =(0)m |W |(0)n E(0)n E(0)m

.

,% $!)$, = *! ',%%"''/1) ") 2%/0 "% ,%4") 4" 2"%$&%-!'/,)"# ! *!# E&)'/,)"#

4" ,)4! #"%5)

(1)n =m 6=n

(0)m |W |(0)n E(0)n E(0)m

(0)m ,

, = *, +&" "# /(&!* = *! E&)'/1) 4" ,)4! ") 2%/0 "% $@%0 /), 4" *! $",%9! 4"

2 "%$&%-!'/,)"# #"%5

n (0)n + (1)n = (0)n +m 6=n

(0)m |W |(0)n E(0)n E(0)m

(0)m .

! #&0 ! 4"* #"(&)4, $@%0 /), ! *! 4"%"'

8

! ',%%" #,-%" $,4,# *,# "#$!4,#

4"

H00"),# "*

n;

!" R #!*'&*"0,# *! ',%%"''/1) ! *! E&)'/1) 4" ,)4! 4"* "#$!4, E&)3

4!0")$!* 4"* 5$,0, 4" 6 4"-/4! ! &)! 2"%$&%-!'/1) '")$%!* 4"* $/2 ,

W = erE0',0, !)$"# = 4,)4"

e"# *! '!%(! 4"* "*"'$%1) ?

E0"# *! !03

2*/$&4 4" &) '!02, "*@'$%/', ; B *

8

!0 /*$,)/!), 4"* #/#$"0 ! 2"%$&%-!4, "#

!

8

,%!

H = H0 + W = H0 + erE04,)4" ',), '"0,# *,# !&$,.!*,%"# ? !&$,E&)'/,)"# 4"

H0R

E(0)n

?

(0)n`m(r)

;

B * "#$!4, E&)4!0 ")$!* 4"

H0"# ), 4"(")"%!4,

(0)100(r)

',) ")"%(9!

E(0)1 =

13,6 "

(0)100(r) =

1pia30

exp[r/a0]. ! $",%9! 4" 2 "%$&%-!'/,)"# 4! *! ',%%"''/1) ! *! E&)'/1) 4" ,)4! 4"*

"#$!4, E&)4!0 ")$!* 4"* 6 ") 2%/0"% ,%4")

(1)1 =

q 6=1,`,m

2pi0

d pi0sin d

0

r2dr (0)q`m(r)(erE0)

(0)100(r)

E(0)1 E(0)q

(0)q`m(r)

= eE0

q 6=1,`,m

2pi0

d pi0sin d

0

dr (0)q`m(r)r

3(0)100(r)

E(0)1 E(0)q

(0)q`m(r).

Hm = (H0 + W )m = Emm.

ki=1

Ci

[Wji E(1)mij

]= 0,

+&" #,)

k2"'&!'/,)"# ',)

k2/)',()/$!#

Ci; B #" #/#$"0 ! $/")" #,*&'/,)"# ),

$%/. /!*"# #/ "* 4"$"%0 /)!)$" 4" *,# ',"A'/")$"#

Ci#" !)&*! = "# 4"'/%

det|Wji E(1)mij | = 0.

B#$, "# *, 0 /#0, +&" 4/!(,)!*/D!% *! 2 "%$&%-!'/1) ") "* "#2!'/, 4" E&)'/,)"#

4" ,)4! 4"(")"%!4!# R

Wji = (0)mj |W |(0)mi; B) *! A(&%!

#" 0&"#$%! "#$, 4"

0!)"%! "#+&"0 5$/'! ; #!4! ")"%(9!

E(0)n

"# !&$,.!*,% 4" 4,# "#$!4,# = ") "#$"

"> "02*, = "$/+&"$!4,# 2,% |n, k(0) ',) k = 1, 2 ; B) 2%/)'/2/, = 2!%! '!*'&*!% *,#!&$,.!*,%"# 4"* 8!0 /$,)/!), 2"%$&%-!4,

H8!-%9! +&" 4/!(,)!*/D!%*, ") *!

-!#" |n, k(0) ',) n = 1, , 2, . . . , ? k = 1, 2 ; ! 2!%$" H0 ?! "# 4/!(,)!* ")"#! -!#" = ")$,)'"# 8!? +&" 4/!(,)!*/D!% *! 2 "%$&%-!'/1)

W; ! !2%,7/0 !'/1)

+&" 4! *! $",%9! 4" 2 "%$&%-!'/,)"# ") 2%/0 "% ,%4") "# 4"#2%"'/!% *,# -*,+&"#

), 4/!(,)!*"# = 4" E,%0 ! +&" 2!%! '!*'&*!% *! ',%%"''/1) !

E(0)n

-!#$! ',)

4/!(,)!*/D!%

W") "* -*,+&" 4" !&$,"#$!4,# +&" ',%%"#2,)4" !

E(0)n

;

E1(0)

1,1 (0)

1,2 (0)

1,1 (0)

1,2 (0)

2,2 (0)

2,1 (0)

2,1 (0)

2,2 (0)

E2(0)

W1,2(1,1)

W2,1(1,1)

W1,1W1,1(1,1)

2,2

W1,1(1,1)

W2,1(1,2) W2,2

(1,2)

W1,1(1,2) W1,2

(1,2)

W2,1(2,1)

W2,2(2,1)

W1,1(2,1)

W1,2(2,1)

W2,1(2,2)

W2,2(2,2)

W1,1(2,2) W1,2

(2,2)

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

%

!" "!0,# "* '!#, #/02*" 4" &) )/."* ',) 4"(")"%!'/1)

; ")"0,#

E&)'/,)"# 4" ,)4!

(0)a1

?

(0)a2

',) /(&!* ")"%(9!

E(0)a

; !%! '!*'&*!% *!# ',%%"'3

'/,)"# ") ")"%(9!

E(1)a1

?

E(1)a2

") 2%/0 "% ,%4") 4" *! $",%9! 4" 2 "%$&%-!'/,)"#

4"-/4!# ! &)! 2"%$&%-!'/1)

W$")"0,# +&" %"#,*."% "* #/#$"0 ! R

C11

[W11 E(1)11

]+ C12W21 = 0

C11W12 + C12

[W22 E(1)22

]= 0

C21

[W11 E(1)11

]+ C22W21 = 0

C21W12 + C22

[W22 E(1)22

]= 0

+&" 2,4"0,# "#'%/-/% ") E,%0 ! 0!$%/'/!*

W11 E(1)11 W21 0 0

W12 W22 E(1)22 0 00 0 W11 E(1)11 W210 0 W12 W22 E(1)22

C11C12C21C22

=

0000

'&?!# #,*&'/,)"# #" ,-$/")") 4"

W11 W21W12 W22 = 0,

+&" ),# 2%,2,%'/,)! *! "'&!'/1)

(W11 ) (W22 )W21W12 = 0.

) 2&)$, /02,%$!)$" "# ),$!% +&" '&!)4, "*/0 /)!0,# *! 2 "%$&%-!'/1) "*

"#$!4, ,-$")/4, "# &)! ',0-/)!'/1) */)"!* 4" *,# "#$!4,# 4"(")"%!4,# !#,3

'/!4,# !* !&$,.!*,%

E(0)m

= "* 2%,-*"0 ! "# +&" ! 2%/,%/ ), #!- "0,# '&5* "# "#!

',0-/)!'/1) ;

H0 = ~2

22 Ze

2

r

? 8"0,# ,-$") /4, *!# !&$,E&)'/,)"#

(0)n (r)

? *,# !&$,.!*,%"#

E(0)n

',)

n = 1, 2, 3, . . .;

B) "* '!#, ") +&"

pc m0c2 2,4"0,# 8!'"% &) 4"#!%%,**, ") #"%/"

T = m0c2

[1 +

1

2

p2

m20c2 1

8

p4

m40c4+ . . . 1

]=

p2

2m20 p

4

8m30c2+ . . .

B* 2%/0 "% $@%0 /), "# "* +&" $")"0,# ") *! "'&!'/1) 4"

E(1)(rel)n` =

1

2c2

[(E(0)n

)2+ 2

Z4e4

a20n3(`+ 1/2)

+ 2E(0)n Z2e2

a0n2

].

%"',%4!)4, *! "72%"#/1) 4"

E(0)n = Z2e2

2n2a0,

,-$")"0,# *! "72%"#/1)

E(1)(rel)n` =

1

2c22

Z4e4

a20n3

[1

`+ 1/2 3

4n

].

+&" 4! *! "72%"#/1) 4" *! ',%%"''/1) %"*!$/. /#$! ! *! ")"%(9! 4"* 5$,0, 4" 6 ")

2%/0 "% ,%4") 4" *! $",%9! 4" 2 "%$&%-!'/,)"# ; ! ")"%(9! 4" &) "#$!4, '!%!'$"3

%/D!4, 2,%

n?

`#"%5

E(0)n +E

(1)(rel)n`

#/ ',)#/4"%!0,# #1*, "#$! ',%%"''/1) ; !

',%%"''/1) %"*!$/. /#$! #/"02%" #"2!%! *,# 4/#$/)$,# .!*,%"# 4"

`2,#/-*"# 2!%!

&) .!*,% 4!4, 4"

n= ? #/"02%" -!> ! *! ")"%(9! ',) %"#2 "'$, !

E(0)n

= ',0,

#" 0&"#$%! ") *! A(&%!

; B # E5'/* ."% +&" *! ',%%"''/1) %"*!$/./#$! #/"02%"

H H + W0 0 rel

Termino relativista

En,l

n

...

l=0

l=n2l=n1

...

n

%

`

%

4! &)! ',)$%/-&'/1) )"(!$/.! ! *! ")"%(9! ; B * $@%0 /), ")$%" ',%'

8

"$"# 4" *!

"'&!'/1)

"# #/"02%" 2,#/$/., ?! +&" 2!%! "* '!#, 0 5# 4"#E!.,%!-*" ") +&"

` = n 1 .!*"1

n 1/2 3

4n> 0.

TS T V Y[ Y ]\X[ ]W\_ XY

"%, *! ',%%"''/1) %"*!$/. /#$! ), "# *! P)/'! +&"

8

!? +&"

8

!'"% ; C * %"!*/D!%

&) $%!$!0 /")$, %"*!$/./#$! 4"* 5$,0, 4" 6 #" ,-$/")" 4" &) 0,4, )!$&%!* &)

$@%0 /), 4" !',2*!0 /")$, "#29)31%-/$! 4" *! E,%0 !

."% !2 @)4/'"

Wso =1

22c2r

dV

dr` s = (r) ` s.

"72%"#/1) +&" .!*" #/"02%" +&"

` 6= 0 = ?! +&" ") "#" '!#, ), 8!? ',%%"''1)"#29)31%-/$! ; B #$! P*$/0 ! "72%"#/1) *! 2,4"0,# "72 */'/$!% !P) 0 5# #/ %"',%3

4!0,# +&" 4"-/4, ! +&"

s = 1/2*,# .!*,%"# +&" 2&"4" $,0!%

j#"%5)

`+1/2?

` 1/2 = ',) *, +&"

E(1)(so)n` =

2 Z4e4

4c2a20n3`(`+ 1/2)(`+ 1)

{

` si j = `+ 1/2(`+ 1) si j = ` 1/2

}.

B#$! "'&!'/1) 0&"#$%! +&" #" E!.,%"'") ")"%(@$/'!0 ")$" *,# "#$!4,# ',)

j =` 1/2 ',) %"#2 "'$, ! *,# j = `+ 1/2 2"%, "# /02,%$!)$" ),$!% +&"

E(1)(so)n` (para j = `+ 1/2)

`=

E(1)(so)n` (para j = ` 1/2)

(`+ 1)

, *, +&" "# /(&!* = A> !4! &)!

`*! /)$"%!''/1) "#29)31%-/$! #"2!%! "#" "#$!4,

")

2j + 1 = 2(` + 1/2) + 1 = 2` + 2"#$!4,#

4"(")"%!4,#

0"),# */(!4,#

+&" "* ,%/( /)!* ") &)! '!)$/4!4

A = B/((` + 1/2)(` + 1))? ")

2j + 1 =2(` 1/2) + 1 = 2` "#$!4,# 4"(")"%!4,# 05# */(!4,# +&" "* ,%/(/)!* ") &)!'!)$/4!4

B = A/(`(`+ 1/2))4" E,%0 ! +&" "* '")$%,/4" 4" *! ")"%(9!

#&0 !

4" $, 4,# *,# 4"#2*!D!0 /")$,# 2%, 4&'/4,# ',) %"#2 "'$, !* "#$!4, ,%/( /)!*

),

.!%9! 4"-/4, ! *! /)$"%!''/1) "#2 9)31%-/$!

2(`+ 1)A

(`+ 1/2)(`+ 1)= 2`

A

`(`+ 1/2)

',0, #" 0&"#$%! ") *! A(&%!

2!%! "* '!#, 4"

` = 1;

Acoplamiento espinorbita

j=l+1/2=3/2 4 estados

j=l1/2=1/2 2 estados

n, l=1 A

2A

H H + W0 0 so

n, ` = 1

%

j

`1/2

`+1/2%

%

T T V X_ \[ Z] Y_ \

C4"0 5# 4" *!# 4,# ',%%"''/,)"# ?! 0 ")'/,)!4!# = ") "* $%!$!0 /")$, %"*!3

$/. /#$! !2!%"'" &) $@%0 /), !4/'/,)!* +&" !'$P! #1*!0 ")$" '&!)4,

` = 0? #"

4"),0 /)! $@%0 /), 4" Q!% /)

WDarwin =pi~2

22c2Ze2(r).

B#$" $@%0 /), 4/'" +&" "* "*"'$%1) ), 2&"4" "#$!% ")

r = 0= "# 4"'/% = ), 2&"4"

"#$!% 4")$%, 4"* 2%,$1) ; ! ./0 ,# +&" #1*!0 ")$" 8!? &)! 2%,-!-/*/4!4 ),

)&*! 4" +&" "* "*"'$%1) "#$@ 4")$%, 4"* )P'*", 2!%! *,# "#$!4,#

s ` = 0;

!%! $")"% ") '&")$! "#$" $@%0 /), &$/*/D!0,# *! $",%9! 4" 2 "%$&%-!'/,)"# ?=

!* /(&!* +&" ") *,# '!#,# !)$"%/,%"# = !0,# *! -!#" |n`s; jm ; ! +&" ,2"%!4,%j2

',)0&$! ',)

(r)2,4"0,# !% *! $",%9! 4" 2 "%$&%-!'/,)"# 2!%! "#$!4,#

), 4"(")"%!4,#

E(1)(D)n` = n`s; jm|WD|n`s; jm =

4piZe2~2

82c2n`s; jm|(r)|n`s; jm.

"%, 2!%!

` = 0?

s = 1/2"* "#$!4, |n 0 1/2; 1/2 m = |n ` = 0 m` =

0 |s = 1/2 ms = m ; #,0, "* ,2 "%!4,% 4"*$! ), !'$P! #,-%" *!# .!%/!-*"#4" "#2 9) ? *! E&)'/1) 4" ,)4! 4" "#2 9) "#$5 ),%0 !*/D!4! ),# +&"4!

E(1)(D)n` = n ` = 0 s = 1/2; j = 1/2 m|WD|n ` = 0 s = 1/2; j = 1/2 m

= s = 1/2 m|s = 1/2 mn ` = 0 m` = 0|WD|n ` = 0 m` = 0

=4piZe2~2

82c2n00|(r)|n00,

2,% *! 4"A)/'/1) 4" *!

"#$" P*$/0, "*"0 ")$, 4" 0!$%/D ), "# 0 5# +&" "*

'&!4%!4, 4" *! E&)'/1) 4" ,)4! ',) )P0"%,# '&5)$/',#

n, ` = 0,m = 0".!*&!4! ") "* ,%/(")

E(1)(D)n` =

4piZe2~2

82c2|n00(0)|2 .

!%! *!# E&)'/,)"#

s= +&" #,) *!# P)/'!# +&" ), #" !)&*!) ")

r = 0= #!- "0,#

+&"

|n`=0m=0(0)|2 = Z3

pin3a30.

,% *, +&" *! ',%%"''/1) +&" 4! "* $@%0 /), 4" Q!% /) "#

E(1)(D)n` =

2 Z4e4

2c2n3a20solo si ` = 0

+&" #1*, !E"'$! ! *,# "#$!4,# ',)

` = 0? *"# !&0")$! */("%!0")$" *! ")"%(9! ;

T T Y ]WX_X_Y \Y Z]

C&+&"

8

"0,# $%!$!4, *!# $%"# ',%%"''/,)"# 0")'/,)!4!# 2,% #"2!%!4, = ")

*! %"!*/4!4 *!# $%"# !'$P!) ',)>&)$!0 ")$" ") "* 5$,0, 4" 6 ; ,% "**, = "* "E"'$,

+&" ,-#"%.!%"0,# "72"%/0 ")$!*0")$" #"%5 *! #&0 ! 4" *,# $%"# ; C "#$! #&0 ! #"

*" 4"),0 /)! &!*0")$" ',%%"''/1) 4" "#$%&'$&%! A)! 4"* 5$,0, 4" 8/4%1("),

Wef = Wrel +Wso +WD,

? *! ',%%"''/1) ") ")"%(9! +&" #" 2%,4&'" #,-%" *! ")"%(9! '!*'&*!4! ") *!

*"''/1) !)$"%/,% #"%9!

E(1)nj (ef) =

2 Z4e4

2c2n3a20

[3

4n 1j + 1/2

].

C ."'"# #" #&"*") !% *!# 4"A)/'/,)"# 4" *! ',)#$!)$" 4" "#$%&'$&%! A)! +&"

"# &)! ',)#$!)$" #/) 4/0 ")#/,)"# =

= e2

~c=

1

137,

? 4" *! ")"%(9! 4"* "#$!4, E&)4!0 ")$!* 4"* 5$,0, 4" 8/4%1("),

E(0)1 =

e2

2a0= 13,6 eV,

4,)4" #"

8

! !4, *! 4"A)/'/1) 4"* %!4/, 4" ,

8

% R

a0 =1

~2

me2.

#,) "**, = *! ',%%"''/1) 4" "#$%&'$&%! A)! #" "#'%/- "

E(1)nj (ef) =

Z4

n3|E(0)1 | 2

[3

4n 1j + 1/2

]< 0,

',) *, +&" *! ")"%(9! 4" *,# )/."*"# 4"* 5$,0, 4" 6 $")/")4, ") '&")$! *!

"#$%&'$&%! A)! 4"2")4" 4"

n?

j? "#

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E2,1 =1

23

(1

137

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2

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= 4,53 105(eV ) 1(cm1)

1,2378 104 (eV ) = 0,365 cm1.

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mer3|`|.

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mer3`.

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dV (r)

dr=

Ze2

r2,

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1

mer

dV (r)

dr`.

v

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pr

p

r

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HM LL NHM

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%,3

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gs = 2= *, +&" #/()/A'! +&"

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gsB/~= "# "* 4,-*" +&" "*

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B/~;

%

! "7/#$")'/! 4"* "#29) 4"* "*"'$%1) "#$5 ',)A%0 !4! 2,% 0&'

8

,# %"#& *$!3

4,# "72"%/0 ")$!*"# = ? #" 0!)/A"#$! 4" E,%0! 0&? 4/%"'$! ") "* "72 "%/0 ")$,

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