! 749
APÉNDICE XI.- LAS FUNCIONES DE ONDA Y DE PROBABILIDAD DE SCHRÖDINGER
!!!!!!!"#!$%&$!'()#*+,$!(-$&$#*.!*$/.%&-'-!*.%!,.%'%0!!"! #$%& %'& ()'*+,%& )+-%.%/& ,0'& %1$01*).%'& 2%& 314/52*.6%/& 0& (0/-*/& 2%& $.0& '*7(,%&
%1$01*8.&2%&).20&'%.)*20,&)&1)'%.)*20,9&
!!"! #$%& ,0& %1$01*8.& 2%& ).20& 2%& 314/52*.6%/& %'& $.& 1).'-/$1-)& 70-%7:-*1)& ;$%& .)&
/%(/%'%.-0&/%0,*202&<='*10&0,6$.0>&(%/)&;$%&,0&%1$01*8.&2%&(/)+0+*,*202&'=&/%(/%'%.-0&
$.0&/%0,*202&<='*109&
!!"#$%#&'()*+,-%#./#%&,/&01(#2%#'(2/#2%#$&3+420(5%+#####!!!1#'!.#*'!$%!2#!/$,'#+%/.!('-'!$3!&-'#%(.-&$!*$!$#$-45'!6!/./$#&.7!"#!4$#$-'38!2#'!.#*'!$%!($-+9*+,'8!&'#&.!$#!$3!$%(',+.!:-$(+&+$#*.!3'!,).6*-$2&2%&).20! ;!,./.!$#!$3!&+$/(.!:-$(+&+$#*.!$3!(%/=)2)! ;7!<.-/'3/$#&$!2%'/.%!$3!%5/=.3.! !('-'!3'!</%1$%.1*08!($-.!('-'!#.!,.#>2#*+-3.!,.#!$3!%5/=.3.! !*$!3'!?$3.,+*'*8!/[email protected]!2%$!$#!$%&$!'()#*+,$!('-'!3'!>-$,2$#,+'!$3!%5/=.3.! 7!!!!!A'!?$3.,+*'*!:#.B-$3'&+?+%&';!$%!3'!*+%&'#,+'!*+?+*+*'!$#&-$!$3!&+$/(.0!:C;!!!! !!!!!D'/=+)#8!(.-!*$>+#+,+9#8!3'!?$3.,+*'*!$%!$3!(-.*2,&.!*$!3'!3.#4+&2*!*$!.#*'!6!3'!>-$,2$#,+'0!
:E;! !
!!!!F$!:C;!6!:E;!%$!*$*2,$!G2$0!!
:H;!! !
!!!!IJ.-'!=+$#8!*$%(2)%!*$!2#!($-5.*.8! 8! 6! 3'!*+%&'#,+'! -$,.--+*'!(.-! 3'!.#*'!$%! 8! *$!/.*.!G2$!3'!:H;!%$!&-'#%>.-/'!$#0!
:K;! !
!!!!A'!>2#,+9#!*$!.#*'!,.%$#.+*'3!$%0!:L;! !!!!!A'!>2#,+9#!%$#.+*'3!$%!3'!/+%/'8!($-.!,.#!2#'!*+>$-$#,+'!*$!>'%$!*$! 0!:M;! !!!!!"#! $3! ,'%.! *$! >2#,+.#$%! %$#.+*'3$%! 6! ,.%$#.+*'3$%8! $3! &+$/(.! *$! 2#! ($-5.*.! $%! !%$42#*.%7!IN'*+$#*.!2#!($-5.*.! 8!3'!>2#,+9#!*'!2#!-$%23&'*.!+*)#&+,.0!:O;! !6!!!:P;! !
!750
!!!!"#$#%&'!(#!)*+!%,'!$-.-/#(0!#1#23-.2&!(#!#%4(3562! 7!)8+! !!!!!9:&/#!$-.-/#(3;#%&'!-(!,.$6(&! 7!)<=+! !!!!!>&.!-'5#!$-.-/#(3;#[email protected]!-(!4-/A&2&!B#!.&!-'! 0!'3.&! !?&%&!?&%4/&C#/-%&'!#!?&.53.6#[email protected]! 6'#.2&! (#'! 32-.532#2-'! 5/3$&.&%D5/3?#'! E6-! #F3/%#.! E6-!
!B!E6-! !B! !
)<<+! !
!!!!!!!! !!!!!G#/#!6.#!&.2#!?&.!(&.$3562! 0!!'-!'3$6-!E6-7!
)<H+ 0!2-!%&2&!E6-!(#!)<=+!'-!E6-2#!?&%&7!
)<I+! !
!!!!>&.!C#'-!-.!)<+!B!)H+0!(#!)<I+!'-!46-2-!-'?/3C3/!5#%C3D.!?&%&7!
)<J+! !
)<K+! !).&5#<LJ<+!!!!!!!!M-#%&'!#:&/#!6.#!$/,F3?#!2-!-'5#!F6.?3@.!?&'-.&32#(!)-.!2&.2-! +7!
!!!!N/#.'F&/%#%&'!#:&/#!(#!-?6#?3@.!)<K+0!5&%#.2&!-.!?6-.5#!E6-!2-'46D'!2-!6.!4-/A&2&! 0!(#! 23'5#.?3#! /-?&//32#! -'! 3$6#(! #! (#! (&.$3562! 2-! &.2#0! -'! 2-?3/0! 0! B! 5&%#.2&! -.! ?6-.5#!5#%C3D.!(#!-?6#?3@.!)H+! 7!
)<O+ !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!<LJ<!9!(#!%3'%#!?&.?(6'[email protected]!4-/&!4&/!&5/&!?#%3.&0!((-$#!9/5:6/!P-3'-/0!!"#$%&'"()*%)+,(-$.)/"*%0#.))))<8OL+7!*I!
! 751
!!!!!!! !"#$%&!'()*+!
!
!!!,-./#/0$1!&2$3&!-4!5#60-3$!7-!$#7&89!
"':+! !
!
!!;!4&!5.3-<=-#</&!&#>=4&389!
"'(+! !
!
!!!?=1%/%=;-#7$!"':+!;!"'(+!-#!"'@+A!$B%-#-0$19!
"'C+! !D! !
!
!!!!!E&!"'C+!-1!!"#$%!"&'("$)*$+")#$,-#"#.$!"')'/*"0'+"#-.$&+"$-+"1'2!)$ $3$%4*&!*"&'#$ 5$
!
!!!,-3/F&0$1!&2$3&!4&!"'C+!7$1!F-<-1!<$#!3-1G-<%$!&! 9!
"*H+! !
!
!!!I!7$1!F-<-1!<$#!3-1G-<%$!&! 9!
"*'+! !
!
!!!J$0B/#$!4&!"*H+!;!4&!"*'+A!$B%-#-0$1!-#$*&!#&'("$)*$+")#$)*$6&748)'"1*49!
"**+!! !
!
!!!E&!1$4=</K#!>-#-3&4!7-!4&!-<=&</K#!"**+A!G&3&!$#7&1!L=-!F/&M&#!-#!4&!7/3-<</K#! !-19!
"*N+! A!-#!7$#7-! !;! A!4&!&0G4/%=7O!!P$3!4$!%&#%$A!4&!.=#</K#!7-!$#7&!7-!
4&!-<=&</K#!"*N+!%/-#-!=#&!G&3%-!<$0G4-M&!;!=#&!G&3%-!3-&4O!!!""#!"$%&'(&))*+!,*''*-*'.,/0$!1&!2.!3*24,/0$!5&$&'.2!1&!2.!&,4.,/0$!1/6&'&$,/.2!!!!!P&3&! <$33$B$3&3! -1%$A! G3/0-3$! 1-G&3&0$1! 4&1! F&3/&B4-1!
"Q+ O!
!!,&7$!L=-!-4!4&7$!/RL=/-37$!-1!/#7-G-#7/-#%-!7-! !;!-4!4&7$!7-3-<2$A!7-! A!1-!1/>=-!L=-!&0B$1!
%S30/#$1!1$#!/#7-G-#7/-#%-1!%&#%$!7-! !<$0$!7-! !;A!G$3!4$!%&#%$A!-L=/F&4-#%-1!&!=#&!<$#1%&#%-!
A!GO-O!!
"QQ+ !E&!1$4=</K#!>-#-3&4!7-!4&!-<=&</K#!"**+A!G&3&!$#7&1!L=-!F/&M&#!-#!4&!7/3-<</K#!
!-19!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!'()*!T!4&!0/10&!<$#<4=1/K#A!G-3$!G$3!$%3$!<&0/#$A!44->&!T3%2=3!U-/1-3A!!"#$%&'"()*%)+,(-$.)/"*%0#.)))"'C@(+9!:)!
!752
"###$!! !
"#%$! !!& '!!!!!!()*)! +,-! .)&)! /-*0120302)2! 2-! 4)! 152)6! 7-! *-+,0-*-! ! &! ! !
"%#$"8$ !!&!!"%##$"9$ '!!!!!!:-! ;*);)! 2-! 217! -3,)3015-7! 20<-*-530)4-7! .1=1>?5-)7! 2-! 7->,521! 1*2-56! 2-! 4)! <1*=)!!
'!@5!-4!3)71!2-!"%#$!!
"%###$! !
"#A$!"B$ !!!!!&!-5!-4!3)71!2-!"%##$!"9$6!7-!70>,-!+,-!
"A$! !!
"A#$!"C$! '!!!!!!D)21!+,-!!"A##$! 6!!!!7-!70>,-!+,-E!"A###$!"F$ !!!!+,-!*-/*-7-5;)!4)!714,30G5!>-5-*)4'!!!!!!H.1*)! ;-5-=17! +,-! -5315;*)*! 417! I)41*-7! 2-! H6! J6! K!L!D'! ()*)! 315203015-7! 05030)4-7! &! 2-!<*15;-*)!)2-3,)2)76!7-!-53,-5;*)6!!"#$!%&#$'(%)*!+%,'%(-.*-!"#!!" !-7!4)!)=/40;,26!+,-!)**0M)!7-!.)!2-70>5)21!315!4)!4-;*)! $E!!"A#%$!(-% +%/'$'%#&0&% + %
!!!!7-!70>,-!+,-!4)!"A#$!"C$!7-!;*)57<1*=)!-5E!
"A%$"N$ '!!!!!@5! 4)! "A%$!7-! ;*);)!2-!,5)!-3,)30G5! 405-)4!315!217! ;?*=0517!2-73151302176!)7O!+,-!/12-=17!2-;-*=05)*!40M*-=-5;-!-4!I)41*!2-!,5)!2-!4)7!217!3157;)5;-7E!"A%#$! !!!!!K1=M05)521!"A###$6!"A#%$!&!"A%#$6!1M;-5-=17E!"A%##$!"P$! !!!!Q1=)521!).1*)!-5!3,-5;)!4)!3152030G5!05030)4! 6!7-!70>,-!+,-E!
"A%###$! !
! 753
!!!!"#$%! &'(! #')'*+,! -)%! '.-%./0)! 1/)'%1! .+)! 2+,! #3$*/)+,! 2',.+)+./2+,! 4! 5+2'*+,! 6/7%$!1/8$'*')#'!-)%!2'!1%,!2+,!.+),#%)#',9!:;<;=! !!!!!!>+*8/)%)2+!1%!:;<<<=?!1%!:;@<=!4!1%!:;<;=?!+8#')'*+,9!:;;=! !!!!!A%!:;;=!',!1%!*/,*%!B-'!1%!:CD=!: =?!EFGFH!!!!!"#$%#&'()*+(#,-#./01%1*2*,%,#3-#,-/*4%#,-#2%#&'()*+(#,-#0(,%#####I-,#/#-/*+,!1%!:C=!4!1%!:JK=!')!1%!:CD=!4!+8#')'*+,9!:CL=! #!!!!M'.+$2'*+,9!:CN=! !!!!4!:CO=! !!!!!I-,#/#-4')2+!:CN=!4!:CO=!')!1%!:CL=?!+8#')'*+,9!!
:CP=! !!!!!M'.+$2'*+,!%Q+$%!1%!/R-%12%2!2'!G-1'$9!:CK=! !!!!!!!I-,#/#-4')2+!1%!:CK=!')!1%!:CP=?!+8#')'*+,9!
:CS=! !!
!!!!T+$!2'6/)/./0)?!1%!'.-%./0)!2'!+)2%!.+)7-R%2%!',!1%!,/R-/')#'9!
:DU=! !
!!!!T+$!2'6/)/./0)!#%*8/3)?!1%!6-)./0)!2'!5$+8%8/1/2%2!',9!
:DJ=! !
!!!!V+*%)2+! ')! .-')#%! B-'! ?! 4! .+*8/)%)2+! 1%! :DU=! 4! :DJ=?! +8#')'*+,! !"#$%&'()&#*+#,-./"/(!(*"*#*+#0'1-2*(&3+-!"
:DC=! ! W!
!754
!!!!!!!!!!!! !
"##$! !!!!!%&! '()&(*+,! "##$! ! -'.-'/',0&! 1&! 2),(*+,! 3'! .-45&5*1*3&3! 3'! 6(7-83*,9'-:!
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
"#H$! !
!!!!!I'(4-3'=4/G!"?J$! !!!!!K'-*D&=4/!1&!'()&(*+,!"?J$!(4,!-'/.'(04!&1!0*'=.4G!
"#L$! !
!!!!M4=5*,&,34!1&!"?J$!A!1&!"#L$>!450','=4/G!
"#N$! ! !
!!!!O74-&!3'-*D&=4/!1&!"?J$!(4,!-'/.'(04!&1!'/.&(*4!"',!1&!3*-'((*+,!3'1!'P'!Q$G!
"#J$! !
!!!!M4=5*,&,34!1&!"?J$!A!1&!"#J$>!450','=4/G!
"#R$! !
!!!!K'-*D&,34!1&!"#R$!(4,!-'/.'(04!&1!'/.&(*4>!450','=4/G!
"#@$!! !
!
!!!M4=5*,&,34!1&!"?J$!A!1&!"#@$>!A!04=&,34!',!()',0&!E)'! >!450','=4/G!
"HB$! ! !
! 755
!!!!"#$%&'#(%)!*+#!,-!#.#&/0-!1%1-,!#)!,-!)+(-!'#!,-!#.#&/0-!$2.312$-!4!,-!#.#&/0-!5%1#.$2-,6!72!,-!5-&10$+,-!)#!(+#8#!#.!,-!'2&#$$29.!'#,!#:#!;<!,-!#$+-$29.!#)=!
>?@A! !
!!!!"#$%&'#(%)!1-(B23.!*+#=!>?CA! !!!!!D%(B2.-.'%!>?@A!4!>?CA<!%B1#.#(%)=!
>?EA! !
!!!!F+,125,2$-(%)!,%)!13&(2.%)!$%.! =!
>??A! !
!!!!7+)121+2(%)!-G%&-!,-!>EHA!4!,-!>?IA!#.!,-!>??A!4!%B1#.#(%)=!
>?JA! !
!!!!K-! #$+-$29.! >?JA! #)! !"! #$%"$&'() *#) +$,-.*&(/#-) *#0#(*&#(1#) *#!) 1Ë!! #.! +.-! )%,-!'2(#.)29.!#)5-$2-,!>#,!#:#!;A6!L.!1&#)!'2(#.)2%.#)!#)5-$2-,#)!)#!8#!-)0=!
>?HA! !
!!!!!M+)*+#(%)!-G%&-! "#!$%&#%'()!*$!+%,-.*')/$-! ')*$0$)*'$)1$!*$"! 1'$2034!$+-.'%! ,-!5-&10$+,-!#)1N!#.!#)1-'%!#)1-$2%.-&2%6!O-&-!#)1%!2.1&%'+$2(%)!+.!.+#8%!)0(B%,%<!-!)-B#&<! <!*+#!&#)#.1-!,-!P+.$29.!'#!%.'-!2.'#5#.'2#.1#!'#,!12#(5%<!'#!(%'%!*+#!,-!>?JA!)#!1&-.)P%&(-!#.!,-!>?QA=!
>?QA!! !
!!R#!>CQA!)#!'#'+$#!*+#!"#$%&'#&()*$+%$,*+#$(*+%-%*+(%*.%$+%"$.(%/-,!#)=!
>?SA! !!!!!R#P2.2(%)!,-!5-&1#!2.'#5#.'2#.1#!'#,!12#(5%!'#!,-!#$+-$29.!>?SA!$%(%!>?TA!>?TA !!!!!7+)121+2(%)!,-!>?TA!#.!,-!>?SA!5-&-!%B1#.#&=!>JIA! !!!!!UG%&-!)+)121+2(%)!,-!>JIA!#.!,-!>?QA<!5-&-!%B1#.#&=!
!756
"#$%! !
!!!!&'!()!"#$%!*'!+'+,-'!.,'/!
"#0%! !
!!!!&'!()!"#0%!*'!+'+,-'/!
"#1%! !
"#2%! !
!!!!3)!"#2%!'*!!"#$%&"%'()#*$#+%,-.*')/$-0#')*$1$)*'$)2$#*$!#2'$3144!%&")*4#!"#1"-25%&!"#$627#$)#$62"*4#$62"%'4)"-'48#####!567-(,*897:!;*<)=6*!)>6?)!'7!-67+8-867'*!@)?)!+)?!,7)!+'A878-897!+'!,7)!A,7-897!+'!67+):!!"#$%&'()&#*+#,&*"#*+#-'./0*(&1+/#+2#%&#',&23/%'3,#4"3+453(',6#7%+#&,#/+8/+2+&3"#%&"#/+"9(*"*#
$:2('"6# "2,'("*"# "# %&"# 8"/3:'%9"# 9(;/+6# *+# +&+/1:"# # <# 4,4+&3,# 6# 7%+# 2+# 4%+=+# ',&#
=+9,'(*"*# 6# +9# '%"96#4%93(89('"*"# ',&# 2%# ',&>%1"*"6# &,2# *"# 9"# $%&'()&# *+# 8/,;";(9(*"*# *+#+&',&3/"/#%&"#8"/3:'%9"#+&#*+3+/4(&"*,#8%&3,#*+9#+28"'(,?3(+48,#@ A6#7%+#2:#/+8/+2+&3"#
%&"#/+"9(*"*#$:2('"B#
Recommended