ФОЭ - Теория, часть 2 (Лекции по ФОЭ), страница 5

DZ®­¨¬ ï ãà ¢­¥­¨¥ ƒ ¬¨«ìâ®­ {Ÿª®¡¨ ª ª ¯à¨¡«¨¦¥­¨¥ ª®à®âª¨å ¢®«­ ¤¥ à®©«ï, ˜à¥¤¨­£¥à ¯à¨èñ« ª § ª«î祭¨î, çâ® ­¥¢à¥¬¥­­ ï ¢®«­®¢ ï äã­ªæ¨ï \¢®«­ ¬ â¥à¨¨ " (x; y; z),ª ª ¯¥à¢®­ ç «ì­® ­ §ë¢ «¨ ¤¥-¡à®©«¥¢áª¨¥ ¢®«­ë, ¤®«¦­  㤮¢«¥â¢®àïâì ãà ¢­¥­¨î: + k2 = 0 ;¢ ª®â®à®¬ ¢®«­®¢®¥ ç¨á«® k = 2 ,   ¤«ï á¯à ¢¥¤«¨¢  ä®à¬ã«  ¤¥à®©«ï:h= h:=mv p’ ª ª ª, ᮣ« á­® ƒ ¬¨«ìâ®­ã, ®¯â¨ç¥áª®¬ã áâண® ¯¥à¨®¤¨ç¥áª®¬ã ¢®«­®¢®¬ã ¯®«î ®¯à¥¤¥«ñ­­®© ç áâ®âë ¢ ®¯â¨ç¥áª¨ ­¥®¤­®à®¤­®©á।¥ ᮮ⢥âáâ¢ãîâ ¤¢¨¦¥­¨ï ¬ â¥à¨ «ì­®© â®çª¨ ¢ ¯®â¥­æ¨ «ì­®¬24¯®«¥ ᨫ á® áâண® 䨪á¨à®¢ ­­®© ¯®«­®© í­¥à£¨¥©, â® ­ ¤® 䨪á¨à®¢ âì ¯®«­ãî í­¥à£¨î ¤¢¨¦¥­¨ï, â ª çâ® ¨¬¥¥¬ ᮮ⭮襭¨ï:p22m + V;pp = 2m (E V ) ;E=â ª¨¬ ®¡à §®¬, ¤«ï ¬¥å ­¨ç¥áª®© ­¥¢à¥¬¥­­®© ¢®«­®¢®© ä㭪樨 ¨¬¥¥¬ á«¥¤ãî饥 §­ ¬¥­¨â®¥ \­¥¢à¥¬¥­­®¥", ¨«¨ áâ æ¨®­ à­®¥ ãà ¢­¥­¨¥˜à¥¤¨­£¥à :2 + 4h2 2m (E V ) = 0 :…᫨ ¢¢¥á⨠¯¥à¥çñભãâãî ¯®áâ®ï­­ãî DZ« ­ª  h = 2h , â® ¯à¨¢¥¤ñ­­®¥ áâ æ¨®­ à­®¥ ãà ¢­¥­¨¥ ¬®¦­® § ¯¨á âì ¢ ¯à¨¢ëç­®¬ ¢¨¤¥:h 22m + V = E :—â®¡ë ¯®«ãç¨âì ¢ ¯à¥¤¥«¥ ª®à®âª¨å ¤¥-¡à®©«¥¢áª¨å ¤«¨­ ¢®«­ ! 0 (¨«¨, çâ® â® á ¬®¥, ¢ ¯à¥¤¥«¥ ¯®áâ®ï­­®© DZ« ­ª , áâ६ï饩áï ª ­ã«î, h ! 0) ¨§ ­¥¢à¥¬¥­­®£® ãà ¢­¥­¨ï ˜à¥¤¨­£¥à  ãà ¢­¥­¨¥ ƒ ¬¨«ìâ®­ {Ÿª®¡¨ ( ­ «®£ ãà ¢­¥­¨ï í©ª®­ «  ¨§ £¥®¬¥âà¨ç¥áª®© ®¯â¨ª¨), ­ ¤® ᤥ« âì á«¥¤ãîéãî § ¬¥­ã ­¥¨§¢¥áâ­®© ä㭪樨:=e S;£¤¥ S { ¤¥©á⢨¥.’ ª¨¬ ®¡à §®¬, ¨¬¥¥¬ á«¥¤ãî騥 ä®à¬ã«ë ᮮ⢥âá⢨ï:k0 S = ä §  (®¯â¨ª )1 S = ä §  (¬¥å ­¨ª ) :hDZ®¢â®àïï, á«®¢®-¢-á«®¢®, ¢ë¢®¤ ãà ¢­¥­¨ï í©ª®­ «  ¨§ ᪠«ïà­®£®­¥¢à¥¬¥­­®£® ¢®«­®¢®£® ãà ¢­¥­¨ï, «¥£ª® ¯®ª § âì, çâ® ¥á«¨ ¨áª âì¯à¨¡«¨¦ñ­­®¥ à¥è¥­¨¥ ãà ¢­¥­¨ï@2@2@2+++ 2hm (E V ) = 022@x@y@z 2¢ ¢¨¤¥(x; y; z) = e S (x;y;z) ;ihih25â® ¯à¨ h ! 0 ¤«ï ­®¢®© ­¥¨§¢¥áâ­®© ä㭪樨 S ¯®«ã稬 á«¥¤ãî饥¯à¨¡«¨¦ñ­­®¥ ãà ¢­¥­¨¥:@S 2@S 2@S 2+ @y + @z = 2m (E V ) ;@xâ.¥.

ãà ¢­¥­¨¥ ƒ ¬¨«ìâ®­  { Ÿª®¡¨.‚ ¢®«­®¢®© ®¯â¨ª¥ ª®íää¨æ¨¥­â, áâ®ï騩 ¯¥à¥¤ ä㭪樥© ¢ ¢®«­®¢®¬ ãà ¢­¥­¨¨, ï¥âáï ª¢ ¤à â®¬ ¢®«­®¢®£® ç¨á« . DZ®í⮬ã ⥯¥àì á«¥¤ã¥â ¯®«®¦¨âì, çâ®2m (E V ) = k2 :h 2’ ª ª ª k = 2 , â® ®âá ­¥¯®á।á⢥­­® á«¥¤ã¥â, çâ®2 =„ «¥¥, EVh22 m (E V ) := T = mv2 2 ,   ¯®â®¬ã 2 m (E=h;mvV ) = mv2 ¨, á«¥¤®¢ â¥«ì­®,á­®¢  ¯®«ãç ¥¬ ä®à¬ã«ã ¤¥ à®©«ï ¤«ï ¤«¨­ë ¢®«­ë ª¢ ­â®¢®© ç áâ¨æë. áᬮâਬ ⥯¥àì ¢à¥¬¥­­ãî ¢®«­®¢ãî äã­ªæ¨î (x; y; z) = e Et (x; y; z) ;äã­ªæ¨ï 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î:@2@22m (E V ) = 0 :@2+++222@x@y@zh 2‘«¥¤®¢ â¥«ì­®, äã­ªæ¨ï 㤮¢«¥â¢®àï¥â á«¥¤ãî饬ã ãà ¢­¥­¨î:h @ h 2 @ 2 @ 2 @ 2 = 2m @x2 + @y2 + @z2 + V (x; y; z) ;i dtª®â®à®¥ ­ §ë¢ ¥âáï ¯®«­ë¬, ¨«¨ ¢à¥¬¥­­ë¬ ãà ¢­¥­¨¥¬ ˜à¥¤¨­£¥à .ih262.6 Š¢ §¨ª« áá¨ç¥áª®¥ ¯à¨¡«¨¦¥­¨¥ ª¢ ­â®¢®© ¬¥å ­¨ª¨.“á«®¢¨ï ª¢ ­â®¢ ­¨ï ®à {‡®¬¬¥à䥫줠.

Š®íää¨æ¨¥­ââã­­¥«ì­®© ¯à®§à ç­®á⨠¯®â¥­æ¨ «ì­®£® ¡ àì¥à ‘ç¨â ï ¯®áâ®ï­­ãî DZ« ­ª  h ¬ «®© ¢¥«¨ç¨­®©, â.¥. áç¨â ï, çâ® ¤¥h ª¢ ­â®¢®© ç áâ¨æë ¬­®£® ¬¥­ì襡ன«¥¢áª ï ¤«¨­  ¢®«­ë = mvå à ªâ¥à­®© ¤«¨­ë l ¨§¬¥­¥­¨ï ¯®â¥­æ¨ «ì­®© í­¥à£¨¨ V (x; y; z) , ¬ë¯à¨å®¤¨¬ ª â ª ­ §ë¢ ¥¬®© ª¢ §¨ª« áá¨ç¥áª®© ª¢ ­â®¢®© ¬¥å ­¨ª¥,â.¥. ª ª¢ ­â®¢®© ¬¥å ­¨ª¥ ¢ ª¢ §¨ª« áá¨ç¥áª®¬ ¯à¨¡«¨¦¥­¨¨.Ž­  ¯®«­®áâìî  ­ «®£¨ç­  £¥®¬¥âà¨ç¥áª®© ®¯â¨ª¥ ¢ ¢®«­®¢®© 䨧¨ç¥áª®© ®¯â¨ª¥.‚ ª¢ §¨ª« áá¨ç¥áª®© ª¢ ­â®¢®© ¬¥å ­¨ª¥ ­¥ ­ã¦­® à¥è âì ¢®«­®¢®¥ ãà ¢­¥­¨¥ ˜à¥¤¨­£¥à , â ª ª ª ¢®«­®¢ ï äã­ªæ¨ï (­¥¢à¥¬¥­­ ï,¨«¨ áâ æ¨®­ à­ ï) ¢ â®çª¥ M ¨¬¥¥â ¢¨¤:S;M =e£¤¥ S - äã­ªæ¨ï ¤¥©á⢨ï, â.¥.

¨­â¥£à « ¡¥àñâáï ¯® ¨á⨭­®© âà ¥ªâ®à¨¨ ¬ â¥à¨ «ì­®© â®çª¨ ª« áá¨ç¥áª®© ¬¥å ­¨ª¨:ZSM =M p dl ;ihMM0Lp£¤¥ p = 2m (E V (x; y; z)) ¨ M0 { ­¥ª®â®à ï ­ ç «ì­ ï â®çª , L {¨á⨭­ ï âà ¥ªâ®à¨ï ᮮ⢥âáâ¢ãî饩 ª« áá¨ç¥áª®© ç áâ¨æë.Ž£à ­¨ç¨¬áï ⥯¥àì à áᬮâ७¨¥¬ ⮫쪮 â ª ­ §ë¢ ¥¬®©, ¢ ª®â®à®© ç áâ¨æ  ¤¢¨¦¥âáï¢ ®¤­®¬ ¨§¬¥à¥­¨¨, â.¥ ¢¤®«ì ¡¥áª®­¥ç­®© ¯àאַ©: 1 < x < +1.DZãáâì, ­ ¯à¨¬¥à, äã­ªæ¨ï ¯®â¥­æ¨ «ì­®© í­¥à£¨¨ V (x) ¨¬¥¥â ¢¨¤, ¨§®¡à ¦ñ­­ë© ­  à¨áã­ª¥.ˆ¬¥¥¬ ¤¢  \¯®â¥­æ¨ «ì­ëå ¡ àì¥à " ¨ ®¤­ã \¯®â¥­æ¨ «ì­ãî ï¬ã"¬¥¦¤ã ­¨¬¨.   à¨áã­ª¥ £®à¨§®­â «ì­®© ¯àאַ©, ¯ à ««¥«ì­®© ®á¨ ¡æ¨áá, ®¡®§­ ç¥­® §­ ç¥­¨¥ § ¤ ­­®© ¯®«­®© í­¥à£¨¨ ç áâ¨æë E . áᬮâਬ ¤¢¨¦¥­¨ï ᮮ⢥âáâ¢ãî饩 ª« áá¨ç¥áª®© â®ç¥ç­®© ç áâ¨æë, â.¥.

¤¢¨¦¥­¨ï, ®¯¨á뢠¥¬ë¥ ª« áá¨ç¥áª®© ¬¥å ­¨ª®©, ª®â®à륢®§¬®¦­ë ¢ 㪠§ ­­®¬ ¤ ­­®¬ ¯®â¥­æ¨ «¥ V (x).‚ ®¤­®¬¥à­®¬ á«ãç ¥ ¢á¥ âà ¥ªâ®à¨¨ ïîâáï, ࠧ㬥¥âáï, ¯àï¬ë¬¨ «¨­¨ï¬¨ ¨ ¬®¦­® «¥£ª® ¯®áâநâì ®¡é¥¥ à¥è¥­¨¥ ãà ¢­¥­¨ï®¤­®¬¥à­®© ª¢ ­â®¢®© ¬¥å ­¨ª¨27ƒ ¬¨«ìâ®­ {Ÿª®¡¨:1 dS 2 + V (s) = E ;2m dxpdS=2m (E V (x) ;dxS (x) = Zxp2m (E V (x) :DZãáâì ¨§ ®âà¨æ â¥«ì­®© ¡¥áª®­¥ç­®á⨠¢ ¯®«®¦¨â¥«ì­®¬ ­ ¯à ¢«¥­¨¨ ®á¨ x ¤¢¨¦¥âáï â®ç¥ç­ ï ç áâ¨æ  á § ¤ ­­®© ¯®«­®© í­¥à£¨¥© E.

DZ® ¬¥à¥ ¯à¨¡«¨¦¥­¨ï ª â®çª¥ a ®­  ¡ã¤¥â ¢áñ ¡®«ìè¥ ¨ ¡®«ìè¥ § ¬¥¤«ïâìáï, ¯®ª  ¢ í⮩ â®çª¥ ­¥ ®áâ ­®¢¨âáï ᮢᥬ (¥ñ ᪮à®áâì ¨ ¨¬¯ã«ìá áâ ­ãâ à ¢­ë¬¨ ­ã«î). ‡ â¥¬, â ªª ª ¢ â®çª¥ a ­  ç áâ¨æã ¤¥©áâ¢ã¥â ᨫ , áâ६ïé ïáï ¤¢¨£ âì¥ñ ¢ ®âà¨æ â¥«ì­®¬ ­ ¯à ¢«¥­¨¨ x, ç áâ¨æ  ¢ë©¤¥â ¨§ á®áâ®ï­¨ï ¯®ª®ï, ­ ç­ñâ ã᪮àïâìáï ¨ ¤¢¨£ âìáï ¢ ®âà¨æ â¥«ì­®¬ ­ ¯à ¢«¥­¨¨ ®á¨ x, ¯®ª  ­¥ 㩤ñâ ¢ ®âà¨æ â¥«ì­ãî ¡¥áª®­¥ç­®áâì.’®çª  x = a ­ §ë¢ ¥âáï \â®çª®© ¢®§¢à â  ¤¢¨¦¥­¨ï". ‚ ª ¦¤ë© ¬®¬¥­â ¢à¥¬¥­¨ ®â१®ª AB ¤ ñâ ¢¥«¨ç¨­ã ª¨­¥â¨ç¥áª®© í­¥à£¨¨ T ª« áá¨ç¥áª®© ç áâ¨æë ¢íâ®â ¬®¬¥­â,   ®â१®ª BC { ¢¥«¨ç¨­ã ¯®â¥­æ¨ «ì­®© í­¥à£¨¨ V ¢ ¤ ­­ë© ¬®¬¥­â¢à¥¬¥­¨.

‚ â®çª¥ x = a ª¨­¥â¨ç¥áª ï í­¥à£¨ï T ®¡à é ¥âáï ¢ ­ã«ì.DZਠx = 1, ­ ®¡®à®â, ¯®â¥­æ¨ «ì­ ï í­¥à£¨ï V ®¡à é ¥âáï ¢ ­ã«ì.DZãáâì ⥯¥àì ª« áá¨ç¥áª ï â®ç¥ç­ ï ç áâ¨æ  ¤¢¨¦¥âáï ­  ãç á⪥®á¨ x ®â â®çª¨ b ¤® â®çª¨ á . Žç¥¢¨¤­® â®çª¨ x = b ¨ x = c ïîâáï\â®çª ¬¨ ¢®§¢à â ", â.¥. â®çª ¬¨ ¯®¢®à®â  ¤¢¨¦¥­¨ï ç áâ¨æë; ç áâ¨æ  ¡ã¤¥â ᮢ¥àè âì ¯¥à¨®¤¨ç¥áª¨¥ ª®«¥¡ ­¨ï, ¤¢¨£ ïáì ¨§ b ¢ á ¨ § â¥¬®¡à â­® ¨§ á ¢ b ¨ â.¤.„¢¨¦¥­¨¥ ç áâ¨æë, ¯à¨ ª®â®à®¬ ®­ ¢áñ ¢à¥¬ï ®áâ ñâáï ¢ ª®­¥ç­®© ®¡« á⨠¯à®áâà ­á⢠, ­ §ë¢ ¥âáï \䨭¨â­ë¬",   ¤¢¨¦¥­¨¥ ¥ñ, ¯à¨ ª®â®à®¬ ®­  ã室¨â ­  ¡¥áª®­¥ç­®áâì, ­ §ë¢ ¥âáï \¨­ä¨­¨â­ë¬".‚ ª¢ ­â®¢®© ¬¥å ­¨ª¥ í­¥à£¨¨ E 䨭¨â­ëå ¤¢¨¦¥­¨© ¬®£ã⠯ਭ¨¬ âì ⮫쪮 áâண® ®¯à¥¤¥«ñ­­ë¥ ¤¨áªà¥â­ë¥ §­ ç¥­¨ï.

ƒ®¢®àïâ, çâ® \í­¥à£¨ï ª¢ ­âã¥âáï "¢ í⮬ á«ãç ¥.28­¥à£¨¨ E ¨­ä¨­¨â­ëå ¤¢¨¦¥­¨© ¢ ª¢ ­â®¢®© ¬¥å ­¨ª¥, ª ª ¨¢ ª« áá¨ç¥áª®© ¬¥å ­¨ª¥, ¬®£ãâ ¡ëâì ª ª¨¬¨-㣮¤­®. ƒ®¢®àïâ, çâ®\í­¥à£¨ï ç áâ¨æë ­¥ ª¢ ­âã¥âáï" ¢ í⮬ á«ãç ¥.DZਬ¥à®¬ ¨­ä¨­¨â­®£® ¤¢¨¦¥­¨ï ï¥âáï ᢮¡®¤­®¥ ¤¢¨¦¥­¨¥ç áâ¨æë, ª®£¤  ­  ­¥ñ ¢®®¡é¥ ­¥ ¤¥©áâ¢ãîâ ­¨ª ª¨¥ ¢­¥è­¨¥ ᨫë( ª®£¤  V 0). ‘¢®¡®¤­®¥ ¤¢¨¦¥­¨¥ ¢ ª¢ ­â®¢®© ¬¥å ­¨ª¥ ­¥ ª¢ ­âã¥âáï.‚뢥¤¥¬ ãá«®¢¨ï ª¢ ­â®¢ ­¨ï ®à {‡®¬¬¥à䥫줠 ¢ ª¢ §¨ª« áá¨ç¥áª®© ª¢ ­â®¢®© ¬¥å ­¨ª¥.DZãáâì ã ­ á ¨¬¥¥âáï ¤¢  §­ ç¥­¨ï ¢®«­®¢®© ä㭪樨 ¢ ¤¢ãå â®çª åx1 ¨ x2 :1=e S ;2=e S : §¤¥«¨¬ ®¤­ã ä®à¬ã«ã ­  ¤àã£ãî ¨ ¯®«ã稬 à ¢¥­á⢮:1 = e (S S )2¨«¨ ä®à¬ã«ã1 = e S 2 ;£¤¥ S { ¯à¨à é¥­¨¥ ª« áá¨ç¥áª®£® ¤¥©áâ¢¨ï ¯à¨ ¯¥à¥å®¤¥ ¨§ â®çª¨x2 ¢ â®çªã x1 , â.¥.ihih1ih122ihS =Zx1p(x) dx ;x2p (x) =p2m (E V (x)) : áᬮâਬ 䨭¨â­®¥ ¤¢¨¦¥­¨¥, ª®£¤  ç áâ¨æ  ª®«¥¡«¥âáï ¬¥¦¤ãâ®çª ¬¨ x = b ¨ x = c.

Žç¥¢¨¤­®0ic = exp @h0Zcbb1p(x) dxAb;12i Z p(x) dxAc = exp @hc  á«¥¤®¢ â¥«ì­®, ¤®«¦­® ¢ë¯®«­ïâìáï à ¢¥­á⢮10Zc2iexp @ h p(x) dxA = 1 ;bc;29¨§ ª®â®à®£® ­¥¯®á।á⢥­­® á«¥¤ã¥â, çâ®c2 Z p(x) dx = 2 n ;hb£¤¥ n = 1; 2; 3 ¨ â. ¤. { 楫®¥ ç¨á«®.„¥©á⢨⥫쭮, ­  ª®¬¯«¥ªá­®© ¯«®áª®á⨠äã­ªæ¨ï ez =1 ⮫쪮¯à¨ z = 2 in.“ç¨â뢠ï, çâ® h = 2h , ¯à¨å®¤¨¬ ª â ª ­ §ë¢ ¥¬®¬ã ãá«®¢¨î ª¢ ­â®¢ ­¨ï ®à {‡®¬¬¥à䥫줠:2Zcp(x) dx = nh ;b£¤¥ ¬­®¦¨â¥«ì 2 ¯à®¨§®èñ« ®â ⮣®, çâ® ãç⥭® ª ª ¯àאַ¥ ( ®â b ªá ), â ª ¨ ®¡à â­®¥ ( ®â á ª b) ¤¢¨¦¥­¨ï. “ª § ­­®¥ ãá«®¢¨¥ ¬®¦­®§ ¯¨á âì ¢ ¢¨¤¥:2Zc p2m (EV (x)) dx = n h ;bâ.¥.

¢ ¢¨¤¥  «£¥¡à ¨ç¥áª®£® âà ­á業¤¥­â­®£® ãà ¢­¥­¨ï, ª®â®à®¥ ¬®¦­® à¥è¨âì ®â­®á¨â¥«ì­® §­ ç¥­¨ï í­¥à£¨¨ E . §à¥è ï íâ® ãà ¢­¥­¨¥ ¯à¨ 䨪á¨à®¢ ­­®¬ n , ¯®«ã稬 E = En.’ ª ¬ë ­ ©¤ñ¬ ¢á¥ ¤¨áªà¥â­ë¥ ª¢ ­â®¢ë¥ í­¥à£¨¨ ¤«ï à áᬠâਢ ¥¬®£® 䨭¨â­®£® ¤¢¨¦¥­¨ï ª¢ ­â®¢®© ç áâ¨æë.2DZਬ¥à. DZãáâì V (x) = fx2 , â.¥. à áᬮâਬ ¯à®á⮩ ®¤­®¬¥à­ë© £ à¬®­¨ç¥áª¨© ®á樫«ïâ®à. ’®£¤  ¨¬¥¥¬á«¥¤ãî饥 ãá«®¢¨¥ ª¢ ­â®¢ ­¨ï:2Zx0s2mEx0fx22dx = n h ;£¤¥ x0 ï¥âáï ª®à­¥¬ ãà ¢­¥­¨ï:fx202 = E;30rá«¥¤®¢ â¥«ì­®, x0 = 2fE . ‘¤¥« ¥¬ ¢ ¨­â¥£à «¥ § ¬¥­ã ¯¥à¥¬¥­­®©:q = 2fE x = xx0 . Žç¥¢¨¤­® ⮣¤ , çâ®fx2f 22E x= 2m (1 2) ;2m (E 2 ) = 2mE 1â ª çâ® ­ è¥ ãá«®¢¨¥ ª¢ ­â®¢ ­¨ï § ¯¨è¥âáï ¢ ¢¨¤¥sZ1pp24 2mE 1 2fE d = n h ;0¨«¨8Ermf1Z p012 = n h :‚¥«¨ç¨­  mf à ¢­  ®¡à â­®© 㣫®¢®© ç áâ®â¥ !10 ª®«¥¡ ­¨© à áᬠâਢ ¥¬®£® ª« áá¨ç¥áª®£® ®¤­®¬¥à­®£® £ à¬®­¨ç¥áª®£® ®á樫«ïâ®à .DZ®ï¢¨¢è¨©áï ®¯à¥¤¥«ñ­­ë© ¨­â¥£à « «¥£ª® ¢ëç¨á«ï¥âáï á ¯®¬®éìî § ¬¥­ë ¯¥à¥¬¥­­®© ¨­â¥£à¨à®¢ ­¨ï: = sin '.

ˆ¬¥¥¬Z2Z2Z1 p21 1221 d = cos 'd' = 2 (1 + cos(2 ')d' = 4 + 4 sin(2') = 4 :0000’ ª¨¬ ®¡à §®¬, ®ª®­ç â¥«ì­® ¨¬¥¥¬ á«¥¤ãî饥 ãá«®¢¨¥ ¤«ï ¢ë¡®à  ¤¨áªà¥â­ëå §­ ç¥­¨© í­¥à£¨¨:8E: !10 : 4 = n h ;¨§ ª®â®à®£® ¯®«ãç ¥¬, çâ®E = En = h !0 n ;§¤¥áì 楫®¥ ç¨á«® n ¯à®¡¥£ ¥â §­ ç¥­¨ï 1; 2; 3 ¨ â.¤. è«¨ í­¥à£¥â¨ç¥áª¨© ᯥªâà ª¢ ­â®¢®£® ®¤­®¬¥à­®£® £ à¬®­¨ç¥áª®£® ®á樫«ïâ®à .

Š ª ¢¨¤¨¬, ®­ íª¢¨¤¨áâ ­â­ë©.qx‚­ãâàì ¯®â¥­æ¨ «ì­®£® ¡ àì¥à , ­ ¯à¨¬¥à, ¢ ®¡« áâì ®â x = a ¤®= b ç áâ¨æ , á࣫ á­® ª« áá¨ç¥áª®© ¬¥å ­¨ª¥, ¯®¯ áâì ­¥ ¬®¦¥â.31„¥©á⢨⥫쭮, ¢ í⮩ ®¡« á⨠¨¬¥¥¬ ­¥à ¢¥­á⢮ E < V (x), ¨§ ª®â®à®£® á«¥¤ã¥â, çâ® T = E V < 0, â.¥. çâ® ª¨­¥â¨ç¥áª ï í­¥à£¨ïç áâ¨æë ®âà¨æ â¥«ì­ ï (  ¬£­®¢¥­­ ï ᪮à®áâì v ¨«¨ ¨¬¯ã«ìá p {ç¨áâ® ¬­¨¬ë¥ ). ‚­ãâਠ¯®â¥­æ¨ «ì­®£® ¡ àì¥à , ¯® ª« áá¨ç¥áª®© ¬¥å ­¨ª¥, ç áâ¨æ  ­¨ª®£¤  ­ å®¤¨âìáï ­¥ ¬®¦¥â.‚ ª¢ §¨ª« áá¨ç¥áª®© ª¢ ­â®¢®© ¬¥å ­¨ª¥ ¤¥«® ®¡á⮨⠨­ ç¥.