Th Hyog Ato Chapt 7
Hyog ato Th vy fist pobl that Schöig hislf tackl with his w wav quatio Poucig th oh s gy lvls a o! lctic pottial gy still plays a ol i a subatoic lvl btw poto a lcto V 4
Schöig q. fo th hyog ato Th pottial ctal foc V ps o th istac btw th poto a lcto. Fo Hyog-lik atos H + o Li ++ Rplac with Z Z: atoic ub. Us appopiat uc ass μ. V 4 V z y x I Catsia cooiats Sphical syty
Schöig q. i pola cooiats Tasfo to sphical pola cooiats bcaus of th aial syty,, 1 1 1 V
Schöig q. fo th hyog ato To b solv with th coitios ψ b oalizabl ψ & its ivativs b cotiuous a gl-valu 4 1 1 1 V
Disio vsus quatu ubs No. of quatu ubs = isio x a paticl i a 3D cub box L x y z Spi quatu ub I oh ol, a cicula obit was foc; Oly ais to b ti. 1D pobl
Spaatio of vaiabls Th wav fuctio ψ is a fuctio of, θ, φ. Th solutio ay b a pouct of th fuctios.,, R Th w ay b abl to spaat th Schöig quatio ito th spaat ifftial quatios, ach pig o o cooiat:, θ, o φ.
Spaatio of vaiabls R R R R R 1 4 4 1 4 R R R R,, R
Solutio of th Schöig quatio Oly a θ appa o th lft si, a oly φ appas o th ight si of th quatio. Th lft si of th quatio caot chag as φ chags. Th ight si caot chag with ith o θ. ach si s to b qual to a costat fo th quatio to b tu. St th costat qual to. Aziuthal quatio costat R R 4
Solutio of th Schöig quatio 1 1 1 4 1 R R Agula quatio Raial quatio 1 1 4 1 4 R R R R costat
quatio fo φ A i Gal solutio Coitio fo gl-valu fuctio A i i 1 A i, 1,, 3,... Magtic quatu ub
quatio fo θ 1 1 x cos x 1 x 1 x 1 x Associat Lg quatio Associat Lg polyoials P l cosθ
quatio fo θ Associat Lg quatio has a solutio, if l = itg a l l : obital quatu ub =, ±1, ±, ±3, ± l Associat Lg polyoials Lg polyoials
quatio fo 1 4 1 R R R 3 ~ 4 ~ ~ 1 ~ ~ ~ 1 4 a Associat Lagu quatio Associat Lagu polyoials
quatio fo Fo a bou stat < 4 3 1 1 1,, 3,... Picipal quatu ub l+1 l -1 l =,1,, 3,, -1
Raial wav fuctios Fist a fw aial wav fuctios R l Subscipts o R spcify th valus of a l.
Sphical haoic fuctios Th solutios fo θ l a φ a lik. Goupig ths solutios togth ito gl fuctios, aft pop oalizatio, fis th sphical haoic fuctios. Y, l sphical haoics
Noaliz sphical haoics
Hyog ato wav fuctios Th aial wav fuctio R a th sphical haoics Y ti th pobability sity fo th vaious quatu stats. Th total wav fuctio ψ ps o, l, a, which bcos l,, R R l l Y l l,
Quatu ubs Th th quatu ubs Picipal quatu ub l Obital agula otu quatu ub Magtic quatu ub Th stictios fo quatu ubs: > l < l Th bouay coitios: = 1,, 3, 4,... Itg l =, 1,, 3,..., 1 Itg l = l, l + 1,...,, 1,..., l 1, l Itg Th hyog ato stats a QUANTIZD!!!
Picipal quatu ub It sults fo th solutio fo R, which ictly flcts th ffct of th pottial gy V. 1 R 4 1 Th sult is that th bou-stat gy is quatiz by 4 3 1 1 1,, 3,... Th lcto gy is quatiz by a cosv.
Obital quatu ub l Th agula otu is quatiz by l a cosv. 1 1 1 1 4 1 4 1 L L L p p K K K V K R obital 1 1 Raial q.
Obital quatu ub l A ctai gy lvl is gat with spct to l c th gy is ipt of l. Us ltt as fo th vaious l valus. l = 1 3 4 5... Ltt = s p f g h... spctoscopic otatio Atoic stats a f to by thi a l. Th bouay coitios qui > l. A stat with = a l = 1 is call a p stat.
Magtic quatu ub L is a vcto quatity. Th associat ictio s to b spcifi. Its ictio bcos ipotat wh a xtal agtic fil is appli, c lcto s L ca b cosi as a sall agtic ipol with its ow -fil aou it. Fo // z, L z = ħ =, ±1, ±,, ±l. y spcifyig th L a L z, th agula otu is fully ti
Why oly L z? what about L x a L y? Quatu chaics allows th agula otu to b quatiz alog oly o ictio i spac. caus of th latio L = L x + L y + L z, th kowlg of two copots woul iply a kowlg of th aiig thi copot. ut this violats th uctaity picipl.
Noal Za ffct I 1896, th Dutch physicist Pit Za show that th spctal lis itt by atos i a agtic fil split ito ultipls. Th Za ffct
Magtic ipol Magtic ot of th obital lcto i a hyog ato Magtic gy of th lcto V s cos cos / / f T L L f T IA L L z V z z z ˆ oh agto μ
Th oal Za ffct Th pottial gy is quatiz. V, 1,..., 1, A sult of spac quatizatio qually spac l+1 sub-gy lvls Wh a agtic fil is appli, th p lvl of atoic hyog is split ito th ifft gy stats. l gy 1 + μ 1 μ
Th oal Za ffct Tasitios fo p to 1s
Th oal Za ffct Tasitios fo to p
Th oal Za ffct A atoic ba of paticls i th l = 1 stat pass though a agtic fil alog th z ictio. Th l = +1 stat will b flct ow, th l = 1 stat up, a th l = stat will b uflct. If th spac quatizatio w u to th agtic quatu ub l, l stats is always o l + 1 a shoul hav pouc a o ub of lis.
St-Glach xpit
lcto spi I 195, Saul Gousit a Gog Uhlbck i Holla popos that th lcto ust hav a itisic agula otu a thfo a agtic ot. I o to xplai xpital ata, th lcto ust hav a itisic spi quatu ub s = ½. Two itisic agula otu stats w cofi fo lcto. y callig th latios btw L, L z, l, a l. s 1 s 1 : spi quatu ub S s s 1 S z s 1 3
Itisic spi Th spiig lcto woul act i a agtic fil siilaly to th obitig lcto. Is it u to th spiig atu of lcto? S I 5S 5 5 3 4 51 ~ 1 17 5 4 c???? Nothlss, Paul Diac fou i his lativistic quatu chaics that th lcto ust hav th itisic agula otu a agtic ot popos by Gousit a Uhlbck 199.
Magtic ipol associat with th itisic spi S V S L S z sz S z sz s Th iffc i th agtic ipol ots fo S a L is a cosquc of thoy of lativity. Gyoagtic atio fo l o s g l = 1 a g s = s s s s z s sz z z g g S g g g L g
Raiativ tasitio Fo a lcto i a gy igstat, th avag positio is Ipt of t!!! Th lcto os ot oscillat. No aiatio!!! x i t / i t / x x x x This cotaicts oh s assuptio o aiatio a also th xpital obsvatio. hf i f i f
QM xplaatio o aiatio <x> has th oscillatio pat. lctic ipol aiatio Ata x b a ab x t i x b a ab x t x x b x x a x x ab x x b a x x b x x a x ab b a b a x b a x b a b a x x x x t i t i / / cos cos i i
QM xplaatio o aiatio Dipol tasitio btw two ig-stats x x Fo hyog ato x : : allow fobi ' l' ' l V wh 1, 1 Slctio uls
gy lvl iaga fo hyog 1, 1 quatu chaics oh ol
lcto pobability sity oh ol Quatu chaics A fiit,θ,φ fo a giv istac A fiit obit No fiit,θ,φ, but oly th lativ pobability fo fiig th lcto No cocpt o obit
Pobability sity W ust us wav fuctios to calculat th pobability istibutios of th lcto. Th positio of th lcto is spa ov spac a is ot wll fi. poit paticl?,,,, R R
Pobability sity Th aziuthal pobability sity is a costat. No pc o φ, galss of, l, A i A i i A cost. Th agula pobability sity ps o l. Y l, Th aial pobability sity ps o a l. R l Ay coctio with th oh ol?
Wav fuctio vs pobability sity
Pobability sity
Pobability R V,,,,,,
Raial pobability sity Th pobability of fiig th lcto btw a + Rgalss of θ a φ,, R P R P R R P l l
Wav fuctio vs aial pobability sity R a P fo th lowst-lyig stats of th hyog ato
Coctio to th oh ol I oh ol a 1s P l = R l I quatu chaics, it cospos to th ost pobabl fo th, l ax = -1 stat. 3 p 4f