Quantum Mechanics Course Qm003
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?- '0 a:DFb+ ?J "ce@G!$d N"#%# P0&(!$'*&;)+@7 - J,.1%KQ!0/(#R -'2'$13/(41S,5/(4 !$/(%?UTW'768VX4 9/Y #% .!2 (&:= /;# -<>#R= 'H?-!2 (4&(?-@ADZ4 =% (!$4&;BC=%4[?E -1SDF/( 4>4>&;41G :/('H4\1 6]4. - -1^ I :6+'H4?J!2P7?J - -1 1 KQK /;DF L4G/;M_` ON r_t fhj$g7k i llHin m oqpsr nRt m r nutwvyx m r nRt '2& i l im o j0k z x f p{r nut}| r nut~vy rj t l2n i g i m Energy V(x)
E Classically allowed region
position
Classically forbidden region
Turning point
-K2=%&(45_2d e&( J/(&!$&;@.#R'$/(41S/( !$?
a O1 /(. I F&; 4>\K2 -'221 CG" 7I7W*7RR z x f psr nut}| j0k vh i r nRt ^ r t g K2i 41 4>&!$?+ ('2?-=+/( -'21IBC'2& r nRt -1/( 4ZBC'H?J?-'0DF J1%K : = H K 2 K \ 4 O / X ; / % 0 ! E / Z D 4 > < $ ! 1 F D ; & L ; / w 4 ; / F 4 m BC'2&;, r nut~v r nutO *¡-¢2£-¤¥ rC¦ t m 4>1 l m v § lH o©¨ r nut lHª ¡J¢H£J¤\¥ r¬ t l2n lHn lHn~« _
l i m v § l i oM¨ lS l2ª o©¨ r nut l i ª ¡-¢2£-¤¥ lHn i Hl n i lHn l2n lHn i « H l H l ª 2 l ª o § lHn ©o ¨ r nut lHn « ¨ l2n \¡-¢H£J¤\¥ v l lHi n oM¨ j lHlH n lHlHªn o©¨ r nut llHni ª f r nutF® lHlHªnw¯ i° ¡-¢H£J¤\¥ b7= % :/( J/(= /( -1 K./(%4"!$R'0i P24&(4\ := ?J/F%!2<²{ J1S/('/; 4.i _`³ a:b c´+DZ49K24/ l lHi n oM¨ j lSl2 n lHlHªn oM¨ r nRt llHni ª f r nutF® lHlHªnZ¯ i o i r nRtO r nRt ° ¡-¢2£-¤¥ v i T 42T-´ i ¶· ¸ ·¹ l l2lHi n lHf ª r nRtF® lHlHlªnZ¯ ª i o i r nRtO r nRtwvº &(4\!$?u/;4>&;, J,.!2K2 -1%!$&;@.#%!$&(/ j lHni lHn o r nRt lHni vº ¼ V½4/\Y e<'21¾ : 6+4>& /; 49 -,5!2K2 -1%!$i &;@5/(4&(,À¿%&; :/d r nRt l2l n ® l2l2ªnZ¯ v f j lHlH n lHlHªn T 42T-´ l l2ª lHn lH® ª l2nw¯ v f j lSl2 n r nRt lHn  l lHr lHª½ªÄÃ0Ã$lHl2n nut v f j lH a1S/(4>KH&;!$/( -1 K/( 49? !2 :/4ÅS=%!$/( -'21K2 -P24 ?-1 ® l2l2ªnw¯ v ?-1 ® Æ r nRt2¯ i '2& lHª v Æ lHn i r nRt Ç3491%'0Dy=% (4È/( 49&;4 (= ?J/ BC'2& lHªÄÃ$l2n -1{/; 49&;4!$?u/;4>&;,É4>)7#%&(4\ ( ( J'H1s/;'KH4/ l i f r nRt §ÊÆ i o r nRtO r nRt~vy lHn i i r nRt « i j
r± t rUµ t r» t rÁ t r _ Ht r _2_ t r_j t r_ t r_¦ t
T 42T-´
l i v r nutÌË § Æ i f r nut0Í l2n i i r nRt « i Z/F/( F#R'2 -1S/Dw4È,.!$²H4Î/; 4"!$#%# &('*)+ -,.!0/( -'21{/(%!$/ li lHn iMÏ DF <?-4!H6 Z/(' §Ð Æ r nRt « i f i r nRt~vy i Â Æ r nut vh r nut i  r nut~v Ò Ñ Æ r nRt Óe ( J1%K5/( F&;4 (= ?L/F -1S/('/( 494>)+# &(4\ ( ( -'21sBC'H& r lHªÄÃ$l2nut K2 -P24\ lHª v Æ lHn Æ Ã$ r nRt v r nRt
r_¬ t r_± t r_µt r_» t r_Á t r j Ht rj _ t
r j2j t  ª r nut~vyÔÉ r nutOlHn ¼ 49K241 4>&!$?Ä ('2?-=+/( -'21'$BÕ/( 4._9`³ a:b cq ´+/(7=% ´ '$B/( 4ÈBC'H&(, rj t m r nRt~vy r nutO *¡J¢H£J¤\¥ Ï Ò Ö r nut *¡Ø×EÙ\£-¤¥Ú¤ DF 4> &;4È4 <'H1% - e6+ J :/('H -'2,1.49BC'H!2&(&~ /; J /(&4e!$P0&;!2@.?J <6+'H J1%/O@. O/'$!$BÄ1S/(/%T 4!2# # &;'*)7 -,.!0/; J'H1 l 9Ã$l2n - Z ( '0DF1 -DF1{ /; ¼ <4BC<'2!2?- (?J4È'0Ö DFDw -4È1 KI%!*D PH!*42@ud da}B p.r nRtwvº /; 4>1 i r nut~vº i F :#¾!0/( !$i ?-?-@5NÌ -1%i 6+Ï 4#¾41%6+41H/\´ -1 r nut~Û *¡ Ùؤ rj ¦ t m T 42T-´ ª r nRtwvºÜn rj ¬ t 4>1
!$1%6 DF <KH JPH4
l2ª vh 2l n lHª i Æ r nRt v lHn vº
rj ± t rj µ t
li v f rj » t l2n i v i i rj Á t Ç34F,.!*@È/;S=¾ <'21¾<?-=%6+4Z/(%!$/Ì/( 4F<'H1%6+ J/( -'21 l "Ã0lHn vº ;!0/( :¿%46I'21 ?-@"DF 41 psr nRt e b+49VÌ# ÝF&;'2Ǻ%!2VS Þß J?- JN~/O@{àÌe6+4>á1¾ÞÎ : JaO/Oâ@.ãäBC= #¾1%'2</(/( -4>'21S1 /; - !2 ?T i i ¼ r Ht å r nutæv ç m r nRt>ç i r _ t inut r Ï 44 DFiÖ J/(/( 4<? !2 ; : <>!2?Ä&(4\ := ?J/; d r nRt F# &;'$N I 4 2 ! ¾ 0 ' H P 4 % # ( & H ' % $ ! ? L O / { @ 6 > 4 % 1 ( L O / @ $ ! H K ( & #R -1 '2&(!/( -&'2!$1%1 !2KH?%4 /;'"lHn /(% 4Î F JPH174>PH?-'74>&<> : L/O4@5?J@.'2# B½&;/('2 #R4'2#%&(/(!2 -&:'2/;1¾ -<>!$?J?R4/(!$'51% J6s/; F/( PH44>?-'7# <>&;'2 L/O%@{!2 J 1[ J?-/( J/O @49'2 -BÄ1S/(¿%41%&(6+P0!$ -1 ? K5l2n!IT #¾!$&(/( <?-4 Oè.@85 ( JI,5 - ?-!2\&I!$2&; K2=%C©,"41Sé}/%´X* JêÌ/IÕ 947!HW :@Q07/('Q¾ (¾ '0Dä/(%!$/I -13/; 4[<>?-!H ( ( <>!$?-?-@NBC'2&; 6 6+41 Z ë &;4>K2 -'21
rm nRt Ò ì r nRt 2í×Ìî £J¤\¥Ú¤+ï ì r nRtEvñð j$k z psr nut f xÈ| r j t Z(5/I5/; 4{¾#RÕ'2 -1S7/;2 "òÏ DFR%4>&;óX4 ¾Ö 0x ô3õÕ% nRtvÉ ´ :#R4<> -!2?E/(&;4!$/(,5g 4>i 1S/I "&;4ÅS= -&;468¾4\<>!2=% :4 !Gr ?-nu L/(t /(?- 4Q I ( J J1 /[KH,5= ? '2!$&;&\4 TA/( 4\< 4s1 D <>!*!2@Q?´Ff '$ Bps= ¾/{r !$/(1%%6+4A?- -1 %K!2 (/( <%4[ -6 (4'2!©?-=+ /( . -'2/;1©%!01 /{4\!$DZ&94A/; %4s!*PH/;4A= &;!M1 J1% ('2KQ?-=+#R/('2 -'2 -1H1W//( -' /( -1H%/;4[4>&;?-#¾4Bö'H/s?-!$!$/(1¾4\6© ~R/(4'3/ODZ/(%4>441s&;/( -K2%S4>/, '2T Ba1{/( '24/( /;= 4&w&(1%DZ J1 '2K8&6 #R ´H'2 - -1s1H/\/;´~ !$4È1¾PS6M <'2 -1 1% J4[/[email protected] '$4B½46 /; 54Î!Q/(= BC'2&;&;1 ,I -1 = K? !#R'2/; -%1S!0// ´ '21%4È!2# # &;'*)+ J,.!0/;4 Ò j0k z x f psr nut}|2à r2 t g i ?-4!26%7 Z@8/('!/( : /(&4È!$BC -'HK2?JS?-'0/"DF?- J -1 1 4{K.'0<P2'24>1%&"1 4!</( ( -,.'21[!$?-BC?E'2 -&;1H,I/;4>= &;? P0!2!$ ?wd !21%6G :'H?JPH4 È/; 4{ a:b+cß4>) !2<Ø/;?J@HT % - ¦
(1)
(2)
E
x=a
Õ -K2= &;4 j dE÷Ì'2/(4>1S/; -!2?R41 4>&;K2@.#%&('2¿%?J4È=% (46[/;'56 4 ;<&; JR4È/( 4"<>'21 1%4<Ø/; J'H1{BC'H&(,= ? !2 T r _ t ëZ!2&(&; -4>&Z/('/(%49&( -K2S/ j <'S 9:Ô]ø r nutOlHn fúù ° û í ì _ í ×7þ ý î £-¤¥Ú¤ t r $ ¦ í ü ¦ Ñ ¤ _ ( J1 :Ô]ø r nutOlHn fúù ° û í ì _ *×7þ ý î £J¤\¥Ú¤ rH¬ t ¦ íü Ñ ¤ r j t ëZ!2&(&; -4>&Z/('/(%49?J4>Bö/ j <'S § Ô ¤ r nRt:l2n f ù û í ì _ í × ý î £J¤\¥Ú¤ r2± t ¦ « í\ü Ñ _ ÿ ¤ rSµ t f Ñ : -1 § Ô ÿ r nRt:l2n f ù¦ « ûí\ü í ì _ × ý î £J¤\¥Ú¤ /( -1 -'21¾! e&; 4> -1Q4"KH J<'H!s'H1[1 &;4>!01 KH/e4\ J<Ø'H (/('21A -,5'2!0149/ÈBC6+ :'H'H &( O,,/!$4= 1%?-6+!H<>4 O//(4>!$'1%1%!2/(<> 49?-4949/(?-'[=%4Bö /(/\ T/('.4'H&( -+K2/;S!2/ J1'$B &;4>/(? %!04"/( -/('2=%1¾&( :1 - -1 #K[¾#¾4>'H/ODw J1S4/È4>1DF L/(/; 4/( : 'H'S?J=+ :4 N ÕS (Õ5ÝÎ3È1%H 4½ '2M B/(2%24S+uò\,5ò R'SM O/.¾ -é~,ò #RÕ'2{&(/;H!2¾1SÕ/# 7&;2'2ò%R?J4],5é} ¾/;0'8ô3DFõÕ% -<È]ò\A/( 24%<2'HõÕ1 1 4\ò<Ø/(Õ -'2A1]ò BC'2{&;,Iò= õÕ? ½!2 !¿%$#%KH= # ¼ &(?J@4 ¾ 4/(?J¾'0D9!0/w´½'2!2B½1%!96#¾/;4 1 454/;4>&;1 !$4/(&( -KH'2@ 1. -'2 ÈBÄ (!"=%#¾<A'2/(/;4>%1S!0/; -/!2?%/( %4!2&(/;&;= J4&(1%&TÕ J1 K #R4e'2¾ -1H!$/&; È&( -4>!$& &;4 w!0 (/ n8'0DF v 15 - 1.!$/(1¾ 6 4 n[v T ¬
x=b
E (1)
(2)
(3)
Õ -K2= &;4 dE= 1 1 4?J -1 K.#R'$/;4>1S/( !$?Ä%!2&(&; -4>&\T x=a
x=b
!$1¾6[¼ (b+'2= ,5# 4#R/('H& (!$4~1%/; (%,5!0 L/Ì/(/(/(4\%6Ä4Z´%#¾ (!$'&(/(/; %<!0?-4/ Õ -!$1 &;4~&(4 -K21% -<'2 16+4>r1S /Ìt BCDZ&;'24È,ßDF/( J%?-?½4Z%?-!*4BöPH/\4 TÌb+'2,54wDF -?-?+¾4Z&(4 %4</(4\6 rm nut v _ ¡ × þ ý Ù\£J¤\¥Ú¤ í ¡ Ñ v _ <>'H § Ô ¤ r nutOlHn f ù¦ « oM¨ _ ( -1 § Ô ¤ r nRt:l2n f ù¦ « r2» t Ñ & w -1%<>?J=%6 ø 46.BC'H&F<'217PH4>1 -4>1%<>4eÑ '2B!$# #%?J@7 -1 ø KI/; 4È<'21%1 4</( -'21sBC'H&:N È 4 # % H ! : e 4 B 2 ! Ø < ; / 2 ' ,I=%?-!H >ad 1&;4>KH J'H1 j t DZ49%!*P24 r ¼
±
_ >< 'H § Ô ¤ r nutOlHn fúù oΨ _ : -1 § Ô ¤ r nRt:l2n fhù ¦« Ñ ø ¦« Ñ ø r2Á t _ _ _ j Ñ ì 4>)7#© f Ô ¤ ø ì r nRt:l2n ° f ¨ Ñ ì 4>)7#© Ô ¤ ø ì r nutOlHn ° T 42T-´ rm nRtwv j_ _ ì 2í× ý þ î £J¤\¥Ú¤ f ¨ _ ì *× ý þ î £-¤¥Ú¤ r¦ Ht Ñ Ñ â'0Dh=% : -1 K i Ôqø ì r nutOlHn v Ô ÿ ì r nRt:l2n o Ôqø ì r nutOlHn _t r ¦ ÿ ¤ ¤ v f Ô ¤ ì r nRt:l2n o r¦ j t ÿ Dw49<!$1 DF&( J/(4 m r nRt !H i rm nRt v j_ _ ì 0×7ý î £J¤¥Ú¤ í f ¨ _ ì í ×7ý î £J¤\¥Ú¤ r¦H t i _ Ñ _ ì 0×7ý î £J¤¥Ú¤ Ñ f ¨ _ ì í ×7ý î £J¤¥Ú¤ t v j Ñ ! 2í Ñ " ! # r 2 ¦ ¦ &; JKHH¼ /\´+eDwKH4È!$ -KH1½4´7/ =% ( J1%K"/(%4<>'21 1 4\<Ø/; J'H1.BC'2&;,I= ? !2 ~BC'2&w/(%4<!2 (4DF 4>&;4/; 4%!$&;&( -4>&~/('/( 4 m$ r nut v j_ H í% _ ( J1 § Ô! ÿ r nRt:l2n fhù¦ « f ¨ j <>'H § Ô ÿ ! r nRt:l2n hf ù¦ « & ¤ ¤ Ñ Ñ r ¦S¬ t _ _ j v j Hí% ( J1 r ªut f ¨ & <'S r ªÄt r ¦H± t Ñ_ _ ¡J¢ í Ñ¡-¢ v j %í f ¨ j f ¨ j' ¡-¢ o j' í ¡-¢ ° r¦7µ t Ñ_ _ v § f ¨ ¦ %í )( ¡J¢ f í ¡-¢ * f ¨ )( ¡J¢ o í ¡J¢+* « r¦H» t Ñ µ
v f ¨ Ñ v f ¨ Ñ 41%<4 -
¶·
§ ¦_ í% ( *¡-¢ f í ¡J¢ * o ( *¡-¢ o í ¡-¢ * « § *¡J¢® o ¦_ í% ¯ o í ¡-¢[® f ¦_ í% ¯ « , , BC'H&(D !$&6s# &;'2#¾!$KH!$/( -1 K %!2<²7D !$&6s# &;'2#%!2KH!$/( -1 K D !*P24 D !*P24
r¦HÁ t rU¬ Ht
·¸ ¹ - . 1%< r nRt~v f ¨ ® & o ¦_ 2í ¯ ¡Ø× ý Ù*£J¤¥Ú¤ í ¡/ m &;4B rU¬ _ t Ñ ¨ ® _ í í ¡ × Ù*£J¤¥Ú¤ í ¡/ u n w t v r fÑ f ¦ ¯ ý m. -e!*P7 -1 K.'2 /;!$ -1 4\6{/; 4"4>)7#%&(4\ ( ( J'H1BC'2& . J1%< r nRt !$1%6 .&(4>B r nRt ´¾DZ4"!$&;491 '0Dú J1A! ¼ #R'H ( L/; J'H1{/;'s<>!$? <=%?-!$/(4/( 4È/;&;!21% (, ; : -'21<'74"m 0s< -4>1S/F=% ( J1%K m Û 2222 m -1% < r nRr nRt t 2222 i 1 jt U r ¬ 2 . 2 2 m r nRt 2 v 222 m -1% < r nRt 222 i t U r $ ¬ 2 . 2 m _ v (U o . í% * rU¬0¦ t i í i 1 Ûv 3 _ o . í i 4 i rU¬2¬ t !$/:BC/;'H4>&F17=%?-!2!0&(/;KH J'H4 1['2B/; 49%!2&(&; J4& 4ÅS=%X!2?-'È Z (/;= ' ,5,5!2&( 5>4H´*BC'2& !Î%!$&;&( -4>&ÕDF J/(I? !$&;K24Z!0/(/(4>17=%!$/( -'21½´$/( 4w/(= 1%1 4>?- J1%K#%&('H%!$ -?- L/O@ í i í i vº í i × þ î £J¤\¥Ú¤ 1 v rU¬$± t (_ o . í * ¼ 49&(4 %4</( -'21 <'7"4 0s<> J41H/ i i Ï v 2222 m J.&(1%4>
v 2222 of .. íí 2222 i rU¬$» t 4" (= ,É'2BX/(%49&(4 %4</( -'21 !$1%6[/(&!$1¾ :,5 ( ( J'H1[<>'S40s< -4>1S/; v ( o _ . %í * o (( of .. %í%í ** i 1 o rU¬$Á t i i v _ o ( i fo i.. o í .7. *6 í i r± Ht i o . o .7. 6 í v i o i. o . í i r± _ t .76 i v _i i r± j t r±2 t  1 o hv _ 9894>/"=%è ÈR1 õ'0ÕDä;<>:½'21%ò+ ( ò\6+S4> &/(%4.# &('H ?-4>, '2B !#R'$/(41S/( !$? DZ4>?-? J18/(%4.Ç=<9ë³!$# #%&('*)+ JN ½ V ,.!0/; J'H1½d V=0
energy
E<0 position (1)
(2)
x=a
(3)
Õ JKH= &(4 ¦ Ìd ëZ'H= 1%6 O/!0/;4#%&('H ?J4, x=b
$P0! !$1¾1 65¼ (BC '2 &(4'H/( &sT?xRÎ4<>/O> <>DZ'24>& 46+1./( J1%%/(K{4s%4e/;#%' ?-!$ J,5&(ÅS/(=% J</ !2?-1S4sn[/(= DFv;, -?J ?U´ /;!$ !21%4><65'H<&('2/;@H& ´Ä6+4 -/(1 ?J% -K,54/( LD /'Q!*nP2<>?-4>;v !HBC= ( ( 1% ´H<><DF!$/( -?E '24/(1Q &(44#R'24>/(&; 1%@244´Õ/(²7&'H!0 J ;1 /;<4>4 -?J/( È? !0< -/;1S44./(1 '%4>!2&;/(K2 <²@4 &;4>K2 -'21[DF 4>&;4 p Wx T a1{/; - &(4K2 -'21½´ !H<><'H&;6 J1 K/('/; 49Ç=<9ëº!$#%# &('*)+ -,.!0/( -'21Xd _ ì í × ý î £J¤\¥Ú¤ r±$¦ t . r nut~v m Ñ Á
¼ 4"!$# #%?J <>!$/( -'21'$BÕ/( 4"<>'21 1 4\<Ø/; J'H1{BC'H&(,= ?-!IBC'2&F&;4>KH J'H1 r j t K2 -P24\ rm nRt~v j <'H § Ô ÿ ¤ r nRtOlHn fúù¦ « i Ñ DF 4>&;4
Ô ¤ r nRt:l2n v Ôqø r nRt:l2n o Ô ¤ r nRtOlHn ÿ ÿ ø ø r nRt:l2n ø Ô R n : t 2 l n Ô r v ÿ ! f ¤ ª ! b7= % :/( J/(= /( -1 K./(%4"!$R'0P24&(4\ := ?J/F J1S/;'5/( 494>)7#%&(4\ ( ( J'H1sBC'H& m r nut Dw4ÈKH4/ i ¤ <>'H § Ô r nutOlHn fúù¦ « v <'HA @ f ª fhù¦B ÿ v <'H @ ª o ù¦ f B v : -1 @ ùj f ª f ù¦ Co B v : -D1 @ ù¦ f ª Co B v : -D1 @ ù¦ f ª B <'S Er t o <>'HF @ ù¦ f ª B ( -1 rG t v : -1 Gr t <>'H @-ª f ù¦ B f <>'H Gr t : -1 @ ª f ù¦ B <4)+'H1 #R1 '24\1 <Ø4/; 1S Z /( /;!$' ?-?J@ ! 4><>)7'21 #R1 '24\1% <Ø 4>/1S Z/( /(!$' ?-?J@ ! 6+ ('24\?-<>=+!*/(@7 -'2 J1%1 K KH ('2&('0?-=+DF/( - -1 '21K r !H<><>4>#+/!$ ?-4 t r 1 '$/e!H<><4#+/;!2 ?-4 t X'54?J -,5 J1%!$/(4È/; 49# 7@+ : <>!2?J?-@s= 1%!H6+,5 - ; ( J ?-49 :'H?J= /( -'21½´+DZ49,I=% :/F%!*P24 v Ô ø r nRtOlHn v r Ij H f _ t ùj ÿ v ® H f j_ ¯ ù 'H& H v _9!$1%6 psr nRt~vy BC'2& J>WKn >L ´+Dw49¾!*P24 Ü r f +tæv ® _ f j_ ¯ v j_ ù _
r±H¬ t r±2± t r±Sµ t r±2» t r±2Á t r}µ Ht r}µ _ t r}µ j t r}µ0 t
r}µ*¦ t r}µ$¬ t r}µ0± t r}µ2µ t
f v j ù NM . v j O ù
f v j ù P i O ¦ v '21 ?-@{ÅS=%!2&:/;4>& DZ!*PH 4> ?- #4>! 1 K2/(% Z1 446[/('5R4 ¿ &(4/:K2/;4 -'261[ -1H J1%/; :'5/(4\/(!2 64"<>'$?-BÕ!H% ( (!$ ?J<>Bö!$N}DZ?-?-@s!*PH!$4>?-?-?J4>'01 DZK24/(6 % BC'2&F -1+¿%1 J/(49¾!$&;&( -4>&
_2_
r}µ0» t r}µ0Á t r» Ht
E Classically allowed region
position
Classically forbidden region
Turning point
-K2=%&(45_2d e&( J/(&!$&;@.#R'$/(41S/( !$?
a O1 /(. I F&; 4>\K2 -'221 CG" 7I7W*7RR z x f psr nut}| j0k vh i r nRt ^ r t g K2i 41 4>&!$?+ ('2?-=+/( -'21IBC'2& r nRt -1/( 4ZBC'H?J?-'0DF J1%K : = H K 2 K \ 4 O / X ; / % 0 ! E / Z D 4 > < $ ! 1 F D ; & L ; / w 4 ; / F 4 m BC'2&;, r nut~v r nutO *¡-¢2£-¤¥ rC¦ t m 4>1 l m v § lH o©¨ r nut lHª ¡J¢H£J¤\¥ r¬ t l2n lHn lHn~« _
l i m v § l i oM¨ lS l2ª o©¨ r nut l i ª ¡-¢2£-¤¥ lHn i Hl n i lHn l2n lHn i « H l H l ª 2 l ª o § lHn ©o ¨ r nut lHn « ¨ l2n \¡-¢H£J¤\¥ v l lHi n oM¨ j lHlH n lHlHªn o©¨ r nut llHni ª f r nutF® lHlHªnw¯ i° ¡-¢H£J¤\¥ b7= % :/( J/(= /( -1 K./(%4"!$R'0i P24&(4\ := ?J/F%!2<²{ J1S/('/; 4.i _`³ a:b c´+DZ49K24/ l lHi n oM¨ j lSl2 n lHlHªn oM¨ r nRt llHni ª f r nutF® lHlHªnZ¯ i o i r nRtO r nRt ° ¡-¢2£-¤¥ v i T 42T-´ i ¶· ¸ ·¹ l l2lHi n lHf ª r nRtF® lHlHlªnZ¯ ª i o i r nRtO r nRtwvº &(4\!$?u/;4>&;, J,.!2K2 -1%!$&;@.#%!$&(/ j lHni lHn o r nRt lHni vº ¼ V½4/\Y e<'21¾ : 6+4>& /; 49 -,5!2K2 -1%!$i &;@5/(4&(,À¿%&; :/d r nRt l2l n ® l2l2ªnZ¯ v f j lHlH n lHlHªn T 42T-´ l l2ª lHn lH® ª l2nw¯ v f j lSl2 n r nRt lHn  l lHr lHª½ªÄÃ0Ã$lHl2n nut v f j lH a1S/(4>KH&;!$/( -1 K/( 49? !2 :/4ÅS=%!$/( -'21K2 -P24 ?-1 ® l2l2ªnw¯ v ?-1 ® Æ r nRt2¯ i '2& lHª v Æ lHn i r nRt Ç3491%'0Dy=% (4È/( 49&;4 (= ?J/ BC'2& lHªÄÃ$l2n -1{/; 49&;4!$?u/;4>&;,É4>)7#%&(4\ ( ( J'H1s/;'KH4/ l i f r nRt §ÊÆ i o r nRtO r nRt~vy lHn i i r nRt « i j
r± t rUµ t r» t rÁ t r _ Ht r _2_ t r_j t r_ t r_¦ t
T 42T-´
l i v r nutÌË § Æ i f r nut0Í l2n i i r nRt « i Z/F/( F#R'2 -1S/Dw4È,.!$²H4Î/; 4"!$#%# &('*)+ -,.!0/( -'21{/(%!$/ li lHn iMÏ DF <?-4!H6 Z/(' §Ð Æ r nRt « i f i r nRt~vy i Â Æ r nut vh r nut i  r nut~v Ò Ñ Æ r nRt Óe ( J1%K5/( F&;4 (= ?L/F -1S/('/( 494>)+# &(4\ ( ( -'21sBC'H& r lHªÄÃ$l2nut K2 -P24\ lHª v Æ lHn Æ Ã$ r nRt v r nRt
r_¬ t r_± t r_µt r_» t r_Á t r j Ht rj _ t
r j2j t  ª r nut~vyÔÉ r nutOlHn ¼ 49K241 4>&!$?Ä ('2?-=+/( -'21'$BÕ/( 4._9`³ a:b cq ´+/(7=% ´ '$B/( 4ÈBC'H&(, rj t m r nRt~vy r nutO *¡J¢H£J¤\¥ Ï Ò Ö r nut *¡Ø×EÙ\£-¤¥Ú¤ DF 4> &;4È4 <'H1% - e6+ J :/('H -'2,1.49BC'H!2&(&~ /; J /(&4e!$P0&;!2@.?J <6+'H J1%/O@. O/'$!$BÄ1S/(/%T 4!2# # &;'*)7 -,.!0/; J'H1 l 9Ã$l2n - Z ( '0DF1 -DF1{ /; ¼ <4BC<'2!2?- (?J4È'0Ö DFDw -4È1 KI%!*D PH!*42@ud da}B p.r nRtwvº /; 4>1 i r nut~vº i F :#¾!0/( !$i ?-?-@5NÌ -1%i 6+Ï 4#¾41%6+41H/\´ -1 r nut~Û *¡ Ùؤ rj ¦ t m T 42T-´ ª r nRtwvºÜn rj ¬ t 4>1
!$1%6 DF <KH JPH4
l2ª vh 2l n lHª i Æ r nRt v lHn vº
rj ± t rj µ t
li v f rj » t l2n i v i i rj Á t Ç34F,.!*@È/;S=¾ <'21¾<?-=%6+4Z/(%!$/Ì/( 4F<'H1%6+ J/( -'21 l "Ã0lHn vº ;!0/( :¿%46I'21 ?-@"DF 41 psr nRt e b+49VÌ# ÝF&;'2Ǻ%!2VS Þß J?- JN~/O@{àÌe6+4>á1¾ÞÎ : JaO/Oâ@.ãäBC= #¾1%'2</(/( -4>'21S1 /; - !2 ?T i i ¼ r Ht å r nutæv ç m r nRt>ç i r _ t inut r Ï 44 DFiÖ J/(/( 4<? !2 ; : <>!2?Ä&(4\ := ?J/; d r nRt F# &;'$N I 4 2 ! ¾ 0 ' H P 4 % # ( & H ' % $ ! ? L O / { @ 6 > 4 % 1 ( L O / @ $ ! H K ( & #R -1 '2&(!/( -&'2!$1%1 !2KH?%4 /;'"lHn /(% 4Î F JPH174>PH?-'74>&<> : L/O4@5?J@.'2# B½&;/('2 #R4'2#%&(/(!2 -&:'2/;1¾ -<>!$?J?R4/(!$'51% J6s/; F/( PH44>?-'7# <>&;'2 L/O%@{!2 J 1[ J?-/( J/O @49'2 -BÄ1S/(¿%41%&(6+P0!$ -1 ? K5l2n!IT #¾!$&(/( <?-4 Oè.@85 ( JI,5 - ?-!2\&I!$2&; K2=%C©,"41Sé}/%´X* JêÌ/IÕ 947!HW :@Q07/('Q¾ (¾ '0Dä/(%!$/I -13/; 4[<>?-!H ( ( <>!$?-?-@NBC'2&; 6 6+41 Z ë &;4>K2 -'21
rm nRt Ò ì r nRt 2í×Ìî £J¤\¥Ú¤+ï ì r nRtEvñð j$k z psr nut f xÈ| r j t Z(5/I5/; 4{¾#RÕ'2 -1S7/;2 "òÏ DFR%4>&;óX4 ¾Ö 0x ô3õÕ% nRtvÉ ´ :#R4<> -!2?E/(&;4!$/(,5g 4>i 1S/I "&;4ÅS= -&;468¾4\<>!2=% :4 !Gr ?-nu L/(t /(?- 4Q I ( J J1 /[KH,5= ? '2!$&;&\4 TA/( 4\< 4s1 D <>!*!2@Q?´Ff '$ Bps= ¾/{r !$/(1%%6+4A?- -1 %K!2 (/( <%4[ -6 (4'2!©?-=+ /( . -'2/;1©%!01 /{4\!$DZ&94A/; %4s!*PH/;4A= &;!M1 J1% ('2KQ?-=+#R/('2 -'2 -1H1W//( -' /( -1H%/;4[4>&;?-#¾4Bö'H/s?-!$!$/(1¾4\6© ~R/(4'3/ODZ/(%4>441s&;/( -K2%S4>/, '2T Ba1{/( '24/( /;= 4&w&(1%DZ J1 '2K8&6 #R ´H'2 - -1s1H/\/;´~ !$4È1¾PS6M <'2 -1 1% J4[/[email protected] '$4B½46 /; 54Î!Q/(= BC'2&;&;1 ,I -1 = K? !#R'2/; -%1S!0// ´ '21%4È!2# # &;'*)+ J,.!0/;4 Ò j0k z x f psr nut}|2à r2 t g i ?-4!26%7 Z@8/('!/( : /(&4È!$BC -'HK2?JS?-'0/"DF?- J -1 1 4{K.'0<P2'24>1%&"1 4!</( ( -,.'21[!$?-BC?E'2 -&;1H,I/;4>= &;? P0!2!$ ?wd !21%6G :'H?JPH4 È/; 4{ a:b+cß4>) !2<Ø/;?J@HT % - ¦
(1)
(2)
E
x=a
Õ -K2= &;4 j dE÷Ì'2/(4>1S/; -!2?R41 4>&;K2@.#%&('2¿%?J4È=% (46[/;'56 4 ;<&; JR4È/( 4"<>'21 1%4<Ø/; J'H1{BC'H&(,= ? !2 T r _ t ëZ!2&(&; -4>&Z/('/(%49&( -K2S/ j <'S 9:Ô]ø r nutOlHn fúù ° û í ì _ í ×7þ ý î £-¤¥Ú¤ t r $ ¦ í ü ¦ Ñ ¤ _ ( J1 :Ô]ø r nutOlHn fúù ° û í ì _ *×7þ ý î £J¤\¥Ú¤ rH¬ t ¦ íü Ñ ¤ r j t ëZ!2&(&; -4>&Z/('/(%49?J4>Bö/ j <'S § Ô ¤ r nRt:l2n f ù û í ì _ í × ý î £J¤\¥Ú¤ r2± t ¦ « í\ü Ñ _ ÿ ¤ rSµ t f Ñ : -1 § Ô ÿ r nRt:l2n f ù¦ « ûí\ü í ì _ × ý î £J¤\¥Ú¤ /( -1 -'21¾! e&; 4> -1Q4"KH J<'H!s'H1[1 &;4>!01 KH/e4\ J<Ø'H (/('21A -,5'2!0149/ÈBC6+ :'H'H &( O,,/!$4= 1%?-6+!H<>4 O//(4>!$'1%1%!2/(<> 49?-4949/(?-'[=%4Bö /(/\ T/('.4'H&( -+K2/;S!2/ J1'$B &;4>/(? %!04"/( -/('2=%1¾&( :1 - -1 #K[¾#¾4>'H/ODw J1S4/È4>1DF L/(/; 4/( : 'H'S?J=+ :4 N ÕS (Õ5ÝÎ3È1%H 4½ '2M B/(2%24S+uò\,5ò R'SM O/.¾ -é~,ò #RÕ'2{&(/;H!2¾1SÕ/# 7&;2'2ò%R?J4],5é} ¾/;0'8ô3DFõÕ% -<È]ò\A/( 24%<2'HõÕ1 1 4\ò<Ø/(Õ -'2A1]ò BC'2{&;,Iò= õÕ? ½!2 !¿%$#%KH= # ¼ &(?J@4 ¾ 4/(?J¾'0D9!0/w´½'2!2B½1%!96#¾/;4 1 454/;4>&;1 !$4/(&( -KH'2@ 1. -'2 ÈBÄ (!"=%#¾<A'2/(/;4>%1S!0/; -/!2?%/( %4!2&(/;&;= J4&(1%&TÕ J1 K #R4e'2¾ -1H!$/&; È&( -4>!$& &;4 w!0 (/ n8'0DF v 15 - 1.!$/(1¾ 6 4 n[v T ¬
x=b
E (1)
(2)
(3)
Õ -K2= &;4 dE= 1 1 4?J -1 K.#R'$/;4>1S/( !$?Ä%!2&(&; -4>&\T x=a
x=b
!$1¾6[¼ (b+'2= ,5# 4#R/('H& (!$4~1%/; (%,5!0 L/Ì/(/(/(4\%6Ä4Z´%#¾ (!$'&(/(/; %<!0?-4/ Õ -!$1 &;4~&(4 -K21% -<'2 16+4>r1S /Ìt BCDZ&;'24È,ßDF/( J%?-?½4Z%?-!*4BöPH/\4 TÌb+'2,54wDF -?-?+¾4Z&(4 %4</(4\6 rm nut v _ ¡ × þ ý Ù\£J¤\¥Ú¤ í ¡ Ñ v _ <>'H § Ô ¤ r nutOlHn f ù¦ « oM¨ _ ( -1 § Ô ¤ r nRt:l2n f ù¦ « r2» t Ñ & w -1%<>?J=%6 ø 46.BC'H&F<'217PH4>1 -4>1%<>4eÑ '2B!$# #%?J@7 -1 ø KI/; 4È<'21%1 4</( -'21sBC'H&:N È 4 # % H ! : e 4 B 2 ! Ø < ; / 2 ' ,I=%?-!H >ad 1&;4>KH J'H1 j t DZ49%!*P24 r ¼
±
_ >< 'H § Ô ¤ r nutOlHn fúù oΨ _ : -1 § Ô ¤ r nRt:l2n fhù ¦« Ñ ø ¦« Ñ ø r2Á t _ _ _ j Ñ ì 4>)7#© f Ô ¤ ø ì r nRt:l2n ° f ¨ Ñ ì 4>)7#© Ô ¤ ø ì r nutOlHn ° T 42T-´ rm nRtwv j_ _ ì 2í× ý þ î £J¤\¥Ú¤ f ¨ _ ì *× ý þ î £-¤¥Ú¤ r¦ Ht Ñ Ñ â'0Dh=% : -1 K i Ôqø ì r nutOlHn v Ô ÿ ì r nRt:l2n o Ôqø ì r nutOlHn _t r ¦ ÿ ¤ ¤ v f Ô ¤ ì r nRt:l2n o r¦ j t ÿ Dw49<!$1 DF&( J/(4 m r nRt !H i rm nRt v j_ _ ì 0×7ý î £J¤¥Ú¤ í f ¨ _ ì í ×7ý î £J¤\¥Ú¤ r¦H t i _ Ñ _ ì 0×7ý î £J¤¥Ú¤ Ñ f ¨ _ ì í ×7ý î £J¤¥Ú¤ t v j Ñ ! 2í Ñ " ! # r 2 ¦ ¦ &; JKHH¼ /\´+eDwKH4È!$ -KH1½4´7/ =% ( J1%K"/(%4<>'21 1 4\<Ø/; J'H1.BC'2&;,I= ? !2 ~BC'2&w/(%4<!2 (4DF 4>&;4/; 4%!$&;&( -4>&~/('/( 4 m$ r nut v j_ H í% _ ( J1 § Ô! ÿ r nRt:l2n fhù¦ « f ¨ j <>'H § Ô ÿ ! r nRt:l2n hf ù¦ « & ¤ ¤ Ñ Ñ r ¦S¬ t _ _ j v j Hí% ( J1 r ªut f ¨ & <'S r ªÄt r ¦H± t Ñ_ _ ¡J¢ í Ñ¡-¢ v j %í f ¨ j f ¨ j' ¡-¢ o j' í ¡-¢ ° r¦7µ t Ñ_ _ v § f ¨ ¦ %í )( ¡J¢ f í ¡-¢ * f ¨ )( ¡J¢ o í ¡J¢+* « r¦H» t Ñ µ
v f ¨ Ñ v f ¨ Ñ 41%<4 -
¶·
§ ¦_ í% ( *¡-¢ f í ¡J¢ * o ( *¡-¢ o í ¡-¢ * « § *¡J¢® o ¦_ í% ¯ o í ¡-¢[® f ¦_ í% ¯ « , , BC'H&(D !$&6s# &;'2#¾!$KH!$/( -1 K %!2<²7D !$&6s# &;'2#%!2KH!$/( -1 K D !*P24 D !*P24
r¦HÁ t rU¬ Ht
·¸ ¹ - . 1%< r nRt~v f ¨ ® & o ¦_ 2í ¯ ¡Ø× ý Ù*£J¤¥Ú¤ í ¡/ m &;4B rU¬ _ t Ñ ¨ ® _ í í ¡ × Ù*£J¤¥Ú¤ í ¡/ u n w t v r fÑ f ¦ ¯ ý m. -e!*P7 -1 K.'2 /;!$ -1 4\6{/; 4"4>)7#%&(4\ ( ( J'H1BC'2& . J1%< r nRt !$1%6 .&(4>B r nRt ´¾DZ4"!$&;491 '0Dú J1A! ¼ #R'H ( L/; J'H1{/;'s<>!$? <=%?-!$/(4/( 4È/;&;!21% (, ; : -'21<'74"m 0s< -4>1S/F=% ( J1%K m Û 2222 m -1% < r nRr nRt t 2222 i 1 jt U r ¬ 2 . 2 2 m r nRt 2 v 222 m -1% < r nRt 222 i t U r $ ¬ 2 . 2 m _ v (U o . í% * rU¬0¦ t i í i 1 Ûv 3 _ o . í i 4 i rU¬2¬ t !$/:BC/;'H4>&F17=%?-!2!0&(/;KH J'H4 1['2B/; 49%!2&(&; J4& 4ÅS=%X!2?-'È Z (/;= ' ,5,5!2&( 5>4H´*BC'2& !Î%!$&;&( -4>&ÕDF J/(I? !$&;K24Z!0/(/(4>17=%!$/( -'21½´$/( 4w/(= 1%1 4>?- J1%K#%&('H%!$ -?- L/O@ í i í i vº í i × þ î £J¤\¥Ú¤ 1 v rU¬$± t (_ o . í * ¼ 49&(4 %4</( -'21 <'7"4 0s<> J41H/ i i Ï v 2222 m J.&(1%4>
v 2222 of .. íí 2222 i rU¬$» t 4" (= ,É'2BX/(%49&(4 %4</( -'21 !$1%6[/(&!$1¾ :,5 ( ( J'H1[<>'S40s< -4>1S/; v ( o _ . %í * o (( of .. %í%í ** i 1 o rU¬$Á t i i v _ o ( i fo i.. o í .7. *6 í i r± Ht i o . o .7. 6 í v i o i. o . í i r± _ t .76 i v _i i r± j t r±2 t  1 o hv _ 9894>/"=%è ÈR1 õ'0ÕDä;<>:½'21%ò+ ( ò\6+S4> &/(%4.# &('H ?-4>, '2B !#R'$/(41S/( !$? DZ4>?-? J18/(%4.Ç=<9ë³!$# #%&('*)+ JN ½ V ,.!0/; J'H1½d V=0
energy
E<0 position (1)
(2)
x=a
(3)
Õ JKH= &(4 ¦ Ìd ëZ'H= 1%6 O/!0/;4#%&('H ?J4, x=b
$P0! !$1¾1 65¼ (BC '2 &(4'H/( &sT?xRÎ4<>/O> <>DZ'24>& 46+1./( J1%%/(K{4s%4e/;#%' ?-!$ J,5&(ÅS/(=% J</ !2?-1S4sn[/(= DFv;, -?J ?U´ /;!$ !21%4><65'H<&('2/;@H& ´Ä6+4 -/(1 ?J% -K,54/( LD /'Q!*nP2<>?-4>;v !HBC= ( ( 1% ´H<><DF!$/( -?E '24/(1Q &(44#R'24>/(&; 1%@244´Õ/(²7&'H!0 J ;1 /;<4>4 -?J/( È? !0< -/;1S44./(1 '%4>!2&;/(K2 <²@4 &;4>K2 -'21[DF 4>&;4 p Wx T a1{/; - &(4K2 -'21½´ !H<><'H&;6 J1 K/('/; 49Ç=<9ëº!$#%# &('*)+ -,.!0/( -'21Xd _ ì í × ý î £J¤\¥Ú¤ r±$¦ t . r nut~v m Ñ Á
¼ 4"!$# #%?J <>!$/( -'21'$BÕ/( 4"<>'21 1 4\<Ø/; J'H1{BC'H&(,= ?-!IBC'2&F&;4>KH J'H1 r j t K2 -P24\ rm nRt~v j <'H § Ô ÿ ¤ r nRtOlHn fúù¦ « i Ñ DF 4>&;4
Ô ¤ r nRt:l2n v Ôqø r nRt:l2n o Ô ¤ r nRtOlHn ÿ ÿ ø ø r nRt:l2n ø Ô R n : t 2 l n Ô r v ÿ ! f ¤ ª ! b7= % :/( J/(= /( -1 K./(%4"!$R'0P24&(4\ := ?J/F J1S/;'5/( 494>)7#%&(4\ ( ( J'H1sBC'H& m r nut Dw4ÈKH4/ i ¤ <>'H § Ô r nutOlHn fúù¦ « v <'HA @ f ª fhù¦B ÿ v <'H @ ª o ù¦ f B v : -1 @ ùj f ª f ù¦ Co B v : -D1 @ ù¦ f ª Co B v : -D1 @ ù¦ f ª B <'S Er t o <>'HF @ ù¦ f ª B ( -1 rG t v : -1 Gr t <>'H @-ª f ù¦ B f <>'H Gr t : -1 @ ª f ù¦ B <4)+'H1 #R1 '24\1 <Ø4/; 1S Z /( /;!$' ?-?J@ ! 4><>)7'21 #R1 '24\1% <Ø 4>/1S Z/( /(!$' ?-?J@ ! 6+ ('24\?-<>=+!*/(@7 -'2 J1%1 K KH ('2&('0?-=+DF/( - -1 '21K r !H<><>4>#+/!$ ?-4 t r 1 '$/e!H<><4#+/;!2 ?-4 t X'54?J -,5 J1%!$/(4È/; 49# 7@+ : <>!2?J?-@s= 1%!H6+,5 - ; ( J ?-49 :'H?J= /( -'21½´+DZ49,I=% :/F%!*P24 v Ô ø r nRtOlHn v r Ij H f _ t ùj ÿ v ® H f j_ ¯ ù 'H& H v _9!$1%6 psr nRt~vy BC'2& J>WKn >L ´+Dw49¾!*P24 Ü r f +tæv ® _ f j_ ¯ v j_ ù _
r±H¬ t r±2± t r±Sµ t r±2» t r±2Á t r}µ Ht r}µ _ t r}µ j t r}µ0 t
r}µ*¦ t r}µ$¬ t r}µ0± t r}µ2µ t
f v j ù NM . v j O ù
f v j ù P i O ¦ v '21 ?-@{ÅS=%!2&:/;4>& DZ!*PH 4> ?- #4>! 1 K2/(% Z1 446[/('5R4 ¿ &(4/:K2/;4 -'261[ -1H J1%/; :'5/(4\/(!2 64"<>'$?-BÕ!H% ( (!$ ?J<>Bö!$N}DZ?-?-@s!*PH!$4>?-?-?J4>'01 DZK24/(6 % BC'2&F -1+¿%1 J/(49¾!$&;&( -4>&
_2_
r}µ0» t r}µ0Á t r» Ht
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