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lnm!P6@;A:!.3%1,W(*&%(.4*X&8(.4&W*&.<!:85!A7!DQEEC9<!NBC@PAE!BC!BC@!:A!:A D A 7A >!#C@!BC!PAD;A!:6D;:QDJ
@C9NA;AD!?87!6E!?CD;A!Y67ECP@AD!C!AE!\A:EA5E!CDEA7B6@!6?!! ≤ ! ≤ !<!?CD;A:!;AE!BA; F! !
!
!!"
!
!
>!
DEN*TIDSUAFHÆNGIGE*SCHRÖDINGERLIGNING
26
*
,A7!?CD;A:!AE!R;E7QP!?87!! ! <!;A7 !R ; A @R P PA D ; A !A 7!6 ?N VD9C9!6?!^ >!,AEEA!A7!CPP A !AE!A9AD E@C9E!\ABC:!?87<!
6E!*ON7`;CD9A7@C9DCD9AD!9V@;A7<!5AD!:D67A7A!AD!R;7A;DCD9<!NB87!;A!ADPA@EA!;A@A!:h@A;A:!\@CBA7!\AJ
:P7ABAE!RD;A7BAM:>!'DA79CAD!C!AE!@RPPAE!:Q:EA5!A7!C!;AD!P@6::C:PA!BA7;AD!9CBAE!BA;!;AD!Y8EADECA@@A!
ADA79C<!!<!89!;AD !P CD AEC:PA!ADA79C<!!
!"#
F!!
! = !
!"#
+ ! =
1
2
!" + !!
!!A7!6?NVD9C9!6?<!NB87!BC!\A?CD;A7!8:t!A@APE7C:P!?A@E<!976BCE6EC8 D :?A @E!A@@A 7!6D;AE<!89!;A75A;!89:h!^>!
)D;:VEEA: !C5 YR @:AD !! = !"!85 :P7ABAE!EC@!!
!
= !
!
!
!
!C!;AD!PCD AEC: PA !AD A7 9C!?h : !!
!"#
=
!
!
!
!
!!
=
!
!
!" =
!
!
!!
>!#C!BV@9A7!6E!;C??A7ADECA7A!Ψ!5A;!NAD:QD!EC@!^<!89!N8@;A7!:h@A;A:!E!P8 D:E6DE>!#C!?h7!;6 F!
!"
!"
= !"!
!"#
= −!"#!
!
!"#
= !!89!BC!?h7!;6!AD !; C??A 7A D EC6@@C9 D CD 9!6?!G>!87;AD>!!!;C??A7ADECA7A:!C9ADF!
!
!
!
!!
!
= !"
!
! = !"
!
! = −!
!
!!
#C!\A5V7PA7<!6E!;AEEA!DAE8Y!:B67A7!EC@!;C??A7ADEC6@@C9DCD9AD!lLm<!NB8 7!;AD !?R@;:EV D ;C9 A!@i:DCD 9!A7!
9CBAE!BA;!! = !
!
cos !" + !
!
sin !" ,!:8 5 !NA7!5h!BV7A!:P7ABAE!BA;!!!
!"#
= 1 ∗ cos !" + ! sin !" >!
,ADDA!P6@;A:!89:h!'R@A7:!?875A@
In
>!!
)!lLm!A7!P!AD!A@@A7!6D; AD !P8 D: E6D E<!5 AD :!BC!N A7!N67!! =
!!
!
!89!! =
!
!
!>!#A;!6E!CD;:VEEA!!!?h:!! =
!!
!
!
=
!
ℏ
<!
NBC@PAE!CD;:VEEA:!Yh!!]:!Y@6; : F!
!
!
!
!!
!
= −
!
!
ℏ
!
!!
,A7!96D9A:!5A;!−ℏ
!
!Yh!\A99A!:C;A7!6?!@C9NA;:EA9DAE<!NB87A?EA7!;A7!;CBC;A7A:!5A;!2!!Yh!\A99A!:C;A7!
89!BC!?h7F! ! ! −
ℏ
!
!!
!
!
!
! !
!
=
!
!
!!
!!
376!EC;@C9A7A!:A:<!6E!;AEEA!DAE8Y!9CBA7!;AD!PCDAEC:PA!AD A79C<!;A7!CD;9h7!C!AD!6D ;AD87;AD:!;C??A7ADEC6@J
@C9DCD9>!, A D !Y 8 EA D ECA @@A!A D A 79 C!A7 !2,]+\.4(4!,]!8<!89!BC!8Y:P7CBA 7!; A D !: 65@A;A!5AP6DC:PA!ADA79C!89!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Ic
!48RD9!lIKKKm<!:>!GIfn!
Ie
!*ON7`;CD9A7!'ZR6EC8D!a!$!:C5Y@A!;A7CBC6EC8D!l@CDPm!
In
!48RD9!lIKKKm<!:>!GIfH!
!
*C;A!OT!6?!"#!
'@ABTTTTTTTTTTTTTTTTTTTTTTTTTTTT! UV7A7TTTTTTTTTTTTTTTTTTTTTTTTTTTT!
!
!
$%&'&%()*+,-.!/01.,*(21!
34!5$6)2%*2*!
!
'@ABD7>FKH!
,6E8F!GH2GIJIKGL!
!
!
782'(&%&8.(.4*9%3:&)8!
'@ABAD:!D6BDF!'556!"789:9667;!
!!!!!!!!!!!!!!!!!!!!!!!!!
"@6::AF!=>:!
!
96D9A7!C9ADDA5!5A;!!
28
F!
Eψ(x) = −
ℏ
!
2!
!
!
ψ x
!!
!
+ !( !)!(!) !!!(8)!
,AEEA!A7!'&.!8('*2,]+\.4(4&!7Z+%^'(.4&%W(4.(.4!Yh!:CD!9ADA7A@@A!?875!C!k D !; C5AD:C8DA!UC9DCD9AD!
P6D!8Y:P7CBA:!Yh!5 6D9A!5h;A7<!5AD!:85!BC!P6D!:A!?76!R;@A; DCD 9AD!:Eh7!;A 7!\@8E!! = !
!"#
+ !!96D9AE!
C9ADDA5!5A;!!>!
,AD!:65@A;A!ADA79C!A7!89:h!\A:P7ABAE!BA;!S65C@E8DJ8YA76E87AD!9CBAE!BA;F!!
! = −
ℏ
!
2!
!
!
!!
!
+ !!(9)!
;6!C5YR@:AD!Y!Q;A7!AD!BC7PD CD9!Yh!?RDPEC8 D AD!BA;!! →
ℏ
!
!
!"
29
>!,AE!?7A59h7<!6E!BC!P 6D !8 Y :P 7CBAF!
!! = !"!
,6!S65C@E8DJ8YA76E87AD!9CBA7!;AD!E8E6@A!ADA79C<!P6D!;AD!\7R9A:!EC@!6E!8Y:P7CBA!AD!:hP6@;E!]3%-&.86
.(.4*-\%'(<!NB87!B C!\ 7 R 9 A7 !ln mF!
! = ! !!
∗
!" = ! !
!
!" = ! ∗ 1 = !!
,A7?87!BC@!S65C@E8D8YA76E87AD!6@EC;!9CBA!'>!
DEN*TIDSAFHÆNGIGE*SCHRÖDINGERLIGNING
30
*
#C!P6D!89:h!BV@9A!6E!;C??A7ADECA7A!Ψ!5A;!NAD:QD!EC@!EC;AD!EF!
!Ψ
!"
= −!"!
!!"#
!
'DA79CAD!A7!9CBAE!BA;!! = !ℏ = ℎ!!89!;A7!96D9A:! C9ADDA5!5A;!ΨF!!Ψ = !ℏΨ>!,A7!96D9A:!5A;!–
!
ℏ
!
Yh!\A99A!:C;A7!6?!@C9NA;:EA9DAE!89!BC!?h7!;6F!−
!
ℏ
!Ψ = −!"Ψ =
!!
!"
>!!Ψ!C:8@A7A:!BA;!6 E!96 D 9A !C9ADDA5!
5A;!−
ℏ
!
!89!BC!?h7F! ! !Ψ = !ℏ
!!
!!
!
,AE!A7!DR!5R@C9E<!6E!CD;:VEEA!R;E7QPPAE!C!lHm<!89!BC!?h7!;AD !8('*, ]+\ . 4 (4&!7Z +%^ ' (.4 &%W(4. (. 4_!
!ℏ
!Ψ
∂t
= −
ℏ
!
2!
!
!
ψ
!!
!
+ !"!!!(10)!
/$-&)"'U!)!'%!GJ,)('%*)0%'U!/ 0 & '%&)$UW- b % , !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
IH
!48RD9!lIKKKm<!:>!GIff!
If
!.7C??CEN:!lGffcm<!:>!II!
=K
!*ON7`;CD9A7!'ZR6EC8D!a!$!:C5Y@A!;A7CBC6EC8D !l@CD P m!
!
*C;A!OV!6?!"#!
'@ABTTTTTTTTTTTTTTTTTTTTTTTTTTTT! UV7A7TTTTTTTTTTTTTTTTTTTTTTTTTTTT!
!
!
$%&'&%()*+,-.!/01.,*(21!
34!5$6)2%*2*!
!
'@ABD7>FKH!
,6E8F!GH2GIJIKGL!
!
!
782'(&%&8.(.4*9%3:&)8!
'@ABAD:!D6BDF!'556!"789:9667;!
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!
#C!\AN6D;@A7!AD!Y67ECPA@!C!AD!GJ;C5AD:C8DA@!RAD;A@C9!Y8EADEC6@\7iD;<!89!\AE7 6 9 EA 7!R ; A @R P P A D ; A!\AJ
BV9A@:A!C!^J6P:AD:!7AEDCD9>!,AD!EC;:R6?NVD9C9A!*ON7`;CD9A7@C9DCD9!
\ADQEEA:<!89!BC!iD:PA7!6E!@i:A!;AD!?87!Y67ECP@AD!C!\7iD;AD>!,AE!9i7A:!
P@67E!NBC@PA!\AECD9A@:A7<!;A7!9V@;A7!?87!Y67ECP@AD<!Dh7!;AD!A7!C!\7iDJ
;AD>!!
!!!!!/67ECP@AD!A7!?R@;:EVD;C9!?7C!C!\7iD;AD<!;A7!?CD;A:!C!CD EA 7B6 @@A E!
0 ≤ ! ≤ !>!,A7!A7!CD9AD!P76?EY hB C7PDCD9<!:h!Y67ECP@AD !7 A ?@A P EA 7A :!
fuldstæn dig *ela st isk!6?!\7iD;AD:!97VD:A7>!,AD!Y8EADECA@@A!ADA79C!
Y67ECP@AD!P6D !6DE69A!A7F!
! ! =
0!!!!å!!!!! ∈]0; ![!
∞!! !!å!!!! ! ∉]0; ![
!
"8D:APBAD:AD!6?!;AEEA!A7<!6E!Y67ECP@AD!6@;7C9!P6D!8YDh!AD!:E87!D8P!ADA79C!EC@!6E!RD;:@CYYA!\7iD;AD<!
?87;C!Y8EADEC6@AE!R;AD?87!A7!RAD;A@C9!:E87E!89!\7iD;AD!A7!RAD;A@C9!;Q\>!,A7?87!:A7!*ON7`;CD9A7@C9DCDJ
9AD!:h@A;A:!R;!?87!Y 67ECP@AD ! CD ; A!C!\ 7i D ;A D F!
Eψ = −
ℏ
!
2!
!
!
ψ
!!
!
!!! (11.1)!
A@@A7!BA;F!
!
!
ψ
!!
!
= −!
!
!!!!(11.2)!
P!A7!9CBAE!BA;!! =
!!"
ℏ
>!Ui:DCD9AD!EC@!lGGm!A7F!
! ! = ! sin !" !! (12)!
NBC@PAE!P6D!\ABC:A:>!,AE!:P6@!9V @;A!?87!\i@9A?RDPEC8DAD<!6E!! = 0!DAE8Y!Dh7!! = 0!89!! = !<!:85!A7!
97VD:ABV7;CA7DA>!387!\i@9A?RDPEC8DAD!9V@;A7!;A!:655A!\AECD9A@:A7!:85!?87!AD!:EhAD;A!\i@9A>!,AD!
?R@;:EVD ; C9 A !@i:D CD 9 !EC@!lGG>Gm!89!lGG>Im!A7!9CBAE!BA;!;AD!P@6::C:PA!?875A@!?87!AD!:C5YA@!N6758DC:P!8:OC@J
@6E87F!! ! = !
!
sin !" + !
!
cos!(!")!MVBD?i7!6?:DCEEAE!85!;C??A7ADEC6@@C9DCD9AD!Q]]u!– !
!
!>!#A;!6E!CD ;J
:VEEA!97VD:A\AECD9 A@:A7DA !?87!Y67ECP@A D !?h7 !BCF!!
! 0 = !
!
sin !0 + !
!
cos !0 = 0 !!!!!!!"!!!!!!! ! = !
!
sin !" + !
!
cos !" = 0!
,AD!ADA:EA!5h;A!6E!! 0 = 0! A7!BA;!6E!@6;A!!
!
= 0<!;6!; A E!?i 7 :E A!@A ; !9 CB A 7!0>!,AE!:P6@!89:h!9V@;A<!6E!
! ! = 0>!
,AE!9V@;A7<!6E!! =
!!
!
>!#C!@6;A7 !!" = !"<!N B 8 7!D !A 7 !AE !N A @E6@!l! = 1,2,3 … )>!!!P6D!:h@A;A:!CPPA!BV7A!0!
A@@A7!6DE69A!AD!DA96ECB!BV 7;C!89!8Y?Q@;A7!! ! = 0<!;6!sin !" !6@EC;!A7!@C9!0<!?87;C!D!A7 !AE !N A@E 6 @!89 !
$(42%!"!`!2&.'&W(4!938&.8(,WX%Y.'!(!a.!
'(1&.*(3.
!
*C;A!OU!6?!"#!
'@ABTTTTTTTTTTTTTTTTTTTTTTTTTTTT! UV7A7TTTTTTTTTTTTTTTTTTTTTTTTTTTT!
!
!
$%&'&%()*+,-.!/01.,*(21!
34!5$6)2%*2*!
!
'@ABD7>FKH!
,6E8F!GH2GIJIKGL!
!
!
782'(&%&8.(.4*9%3:&)8!
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!
sin ! = 0>!#C!@6 ; A7 !!
!
= !>!,A7?87!?h:<!6E !lGIm !DA E8 Y !A7!@i :D CD 9 AD
=G
>!!!P6D!:h@A;A:!:P7CBA:F!!
!
!
=
!"
!
!!
*h@A;A:!P6D!Y67ECP@AD!6DE69A!D89@A!8(WW,'8&!ADA79CEC@:E6D;A<!NB87EC@!;A7!Ni7A7!AD!\A:EA5E!\i@9A?RDPJ
EC8D<!89!PR D !; C :: A!E C@:E 6D;A!A7!5 R @C9 A !?8 7!; AD !6E!\A?CD;A!:C9!C>!'DA79CAD!C:8@A7A:F!
! =
2!"
ℏ
⇔ !ℏ = 2!" ⇔ !ℏ
!
= 2!" ⇔ ! =
!
!
ℏ
!
2!
!
+;E7QPPAE!?87!!
!
!CD;:V EEA :!Yh !P]:!Y @6; :<!; 6!' !89 :h !\A EA9 D A7 !Y6 7ECP @AD :!A DA 79 C!BA; !D F!
!
!
=
(
!"
!
!)
!
ℏ
!
2!
=
!
!
!
!
ℏ
!
!
!
2!
!
'DA79CAD!Y67ECP@AD!P6D!6DE69A!A7!;6!9CBAE!BA;F!
32
!
!
!
=
!
!
!
!
ℎ
!
2!!
!
!
'D!ADA79CEC@:E6D;!A7!:h@A;A:!*8,8(3.\%<!;6!A D A7 9 CA D!CPPA!VD;7A7!:C9!C!;AD!ADP A@EA !EC@:E6 D;!a!89!;A7J
5A;!NA@@A7!CPPA!C5YR@:AD<!;6!Y67ECP@AD:!ADA79C!R;A@RPPAD;A!A7!9CBAE!BA;!;AD!PCDAEC:PA!ADA79C>!! ≠ 0!
;6!;AE!:h!BC@@A!\AEQ;A<!6E!! = 0!89!;A75A;!89:h!! = 0>!,AEEA!\AEQ;A7<!6E!;AD!mindste*energi*Y67ECP@AD!
P6D!6DE69A!A7!BA;!! = 1!89!;A75A;!!
!
=
!
!
!
!
!! !
!
>!,ADDA!P6@;A:!.2W92.)8*&.&%4(&.>!,ADDA!PB6DEC:AJ
7CD9!6?!ADA79C!:E7C;A7!C58;!;AD!P@6::C:PA!5AP6DCP>!
#C!BA;<!6E!\i@9A?RDPEC8DAD!A7!9CBAE!BA;!! = !sin!( !")<!89!BC!iD: PA 7 !6E !P AD ; A !$<!:85!89 : h!P 6 @; A :!
.3%1&%(.4*)3.*8,.8&.
;;
>!#C!P6D !\ ADQEEA!5A E8 ; A D !P 6@;AE!.3%1&%(.4
;G
<!NB87!.3%1&%(.4*X&8(.6
4&W*&.!\7R9A:<!:85!BC:A7!:C9!Q;A7:E!DQEEC9!EC@!?875h@AE>!*6D;:QD@C9NA;AD!?87!6E!?CD ;A!Y67ECP@AD!C!CDEA7J
B6@@AE!0 ≤ ! ≤ !!A7!1<!NB C@P A E!P 6 D !8 Y:P7CBA:!:85F!!
!"#$ !"
!
!
!
!" = 1!
+;E7QPPAE!?87!!!CD;: V EEA :F!
!
!
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!
!"
!
!
!
!
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-A9DA7A9@AD!C!;AD!97h!P6::A!EC@!NiM7A!\7R 9A :!8 9 !B C!?h7 F!
!!!!!!!!!!!! !
!
∗
1
2
1 − !"# 2
!"
!
!
!
!
!" = 1!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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!.7C??CEN:!lGffcm<!:>!IcJIe!
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!48RD9!lIKKKm<!:>!GIfc!
==
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=L
!48RD9!lIKKKm<!:>!GIfn!
sin
!
(
!
)
=
1
2
(1 − cos
(
2!
)
)
!
!
*C;A!O#!6?!"#!
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!
!
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'@ABD7>FKH!
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!
782'(&%&8.(.4*9%3:&)8!
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,A7!?87AE69A:!;6!AD!(.8&4%, 8(3 . !-& ' !*2 X *8(82 8(3 . !?8 7!6 E!@i :A !CDEA 9 76@AEF!
,AD!CD;7A!?RDPEC8D!A7!9CBAE!BA;F!! ! = 2
!"
!
!!
!!;C??A7ADECA7A:!NB87A?EA7!!"!C:8@A7A:!?87! 6E!?h!AE!R;E7QP!?87!!"F!! !
!"
!"
= 2
!"
!
⇔ !" =
!
2!"
!" !
!
!
!89!
!
!
!96D9A:!Yh!R;AD?87!CDEA976@AE!89!R;E7QPPAE!?87!!"!CD; :V EEA :F!
!
!
∗
1
2
∗ 1 − !"# !
!
2!"
!
!
!" = 1!
!
!!"
!P6D!Q;A7@C9A7A!96D9A:!Yh!R;AD?87!CDEA976@AE!89!;A7!CDEA97A7A:F!
1
2
∗ !
!
∗
!
2!"
∗ !
!
!
− sin !
!
!
= 1!
U!P6D!:P7CBA:!:85!! = 2!" !89!BC!N67!;6F!
!
!
1
2
!
2!"
∗ !
!
!!"
− sin !
!
!!"
= 1 ⇔ !
!
1
2
!
2!"
2!" − 0 − sin 2!" − sin 0 = 1!
+;E7QPPAE!7A;ROA7A:F!
!
!
1
2
!
2!"
2!" − 0 − 0 = 1 ⇔ !
!
1
2
!
2!"
2!" = 1!
!
!
!
!
= 1!
#C!N67!; 6!!
!
=
!
!
⇔ ! =
!
!
<!NB87!BC!DiMA:!5A ; !6E !8 Y :P 7C BA !; A D !Y 8 :CE CB A!7 A A@@A !@i : D CD 9 !?8 7!$>!*65E@C9 A !
@i:DCD9A 7!?8 7 !Y6 7E CP @AD !C!\ 7i D ; A D !P 6D !AD; A@C9E!8Y:P7CBA:F!
!
!
(!) =
2
!
sin!(
!"
!
!)!
89!ADA79CAD!A7!9CBAE!BA;F!
!
!
=
!
!
!
!
ℎ
!
2!!
!
!
Ui:DCD9AD!BC:A7<!6E!;A7!A7!AE!RAD;A@C9E!6DE6@!@i:J
DCD9A7<!;6!D!P6D!6DE69A!6@@A!NA@E6@>!3C9R7!=!l6m!
C@@R:E7A7A7!;A!E7A!?i7:EA!@i:DCD9A7!9CBAE!BA;!NADJ
N8@;:BC:!!
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!
!
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!
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