Hydrogen-like Atoms. We can separate the center of mass motion from the internal motion by the transformation

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1 Hydogn-ik Atos A hydogn ik ato consists of on nucus of chag Z and a sing cton of chag -. Th cassica ngy of this syst is th su of th kintic ngis of both patics and thi Couobic attaction' Z E = N & N+ & 4πε N Figu showing position vctos ativ to abitay oigin. W can spaat th cnt of ass otion fo th intna otion by th tansfoation = and n R = + NN / t wh t = + n. Fo ths w gt N = R & = R+ t N t and th ngy bcos Z E = R & t + μ & 4πε N wh μ = + N is th ducd ass. Th cnt of ass ngy is Ec = R = otion. Th ngy du to th ativ otion of th cton and nucus is E & Pc t wh P c is th ina ontu associatd with th cnt of ass t Z p Z int = μ & = 4πε μ 4πε Wh p is th ina ontu associatd with th ativ otion. Th Haitonian fo th two patics is thn J. F. Haison Michigan Stat Univsity

2 ˆ ˆ ˆ ˆ ˆ Pc p Z H = Hc + Hint = + μ 4πε t Wh th cassica onta hav bn pacd by th quantu chanica opatos ˆ P c =ih and ˆp =ih c Th Schoding quation fo th two patic syst is thn ĤΨ= EΨ and bcaus th cnt of ass otion is indpndnt of th intna otion w can wit th wavfunction fo th two patic syst as a poduct Ψ ( R, = φ( R ψ and th tota ngy E can b wittn as th su of th cnt of ass ngy and th intna ngy, E E +E wh = c int ˆ ˆ H φ R = E φ R & H ψ = E ψ and c c c c int int int int ˆ h h Z H = & H = 4πε ˆ c c int t μ In th absnc of an xtna fid, th cnt of ass otion is that of a nuta f patic. If th syst is in f spac (not boundd th wav function is a pan wav and th ngy a continuous function of th patics ontu. ik R h k φ ( R, k = C & Ec = t wh < k <+ Th ngy (in th SI syst associatd with th intna otion is found by soving th ignvau quation Hˆ h Z ψ = ψ = Eintψ μ 4πε int Subjct to th bounday conditions that ψ b finit at th oigin and vanish at. This is don in dtai in any txts and w wi sktch th pocdu and discuss th suts. Not that athough th ngy of th ato is th su of th ngy associatd with th cnt of ass otion and th intna otion on aways assus that th ato is at st J. F. Haison Michigan Stat Univsity

3 and th cnt of ass ngy is zo. W wi ca th intna ngy sipy th ngy and Hˆ Hˆ. int Th kintic ngy opato in sphica poa coodinats is h h h = sinθ + μ μ μ sinθ θ θ sin θ φ And sinc th squa of th obita angua ontu opato ˆL is givn by Lˆ =h sinθ + sinθ θ θ sin θ φ this ay b wittn as h h L = μ μ μ ˆ + Th fist t on th ight is th adia kintic ngy opato whi th scond t is th angua kintic ngy opato. Th ignfunctions of ˆL a w know to b th sphica haonics, Y ( θ, φ with ignvau + ( h ˆ ( θ, φ = ( + h ( θ, φ LY Y Wh is th angua ontu quantu nub and is a positiv intg, < and is th agntic quantu nub and fo a givn is an intg. If w sk a soution to h ˆ L Z (,, E (,, + ψ θφ ψ θφ = μ μ 4πε And wit ψ as a poduct of a adia function and a sphica haaonic, as ψ = R ( Y ( θφ, w find that th adia pat of th wavfunction and th intna ngy a dtind by th diffntia quation h ( + h Z R( ( + = ER μ μ 4πε J. F. Haison Michigan Stat Univsity

4 Lts fist ook at th sphica haonics. Angua Functions Th xpicit fo of Y ( θ, φ is Y with ( θ ( φ ( θ, φ =Θ Φ ( φ Φ = π i φ and ( + ( ( +!! Θ ( θ = δ P θ ( cos P (cosθ is an associatd Lgnd poynoia which ay b wittn in ts of th Lgnd poynoias P ( x as d P ( x = ( P x x dx Th fist fiv Lgnd poynoias a coctd bow. P( x = P( x = x P ( x = (x P ( x = (5x x P x x x = (5 + δ in th dfinition of Θ ( θ is a phas facto which is not univsay agd upon. W wi choos it to b ( whn > and + whn <. This is oftn cad th Condon-Shoty choic of phas and is sotis wittn as th quint J. F. Haison Michigan Stat Univsity 4

5 ( that Y = ( Y *. Not that th z coponnt of th obita angua ontu Lˆz =ih dpnds ony on φ and whn it acts on Y ( θ, φ ony affcts φ i φ Φ φ = so Y ( θ, φ is aso an ignfunction of L ˆz with ignvau h π ˆ LY ( θ, φ = hy ( θ, φ. z Th fist fw sphica haonics a givn bow aong with thi Catsian psntation obtaind using x= sinθ cosφ y = cosθ sinφ z = cosθ Its coon to idntify an obita angua ontu with a tt cosponding to th obita angua ontu quantu nub. This convntion is of histoica oigin and fs to th natu of th spctoscopic ins in th hydogn ato. Th fist fou w cad shap, piay, diffus and fundanta. Th aining tts continu in squnc with j bing oittd tt s p d f g h i s obitas, = p obitas, = d obitas, = Y = Y 4π z = cosθ = 4π 4π i x iy φ + = = 8π 8π ± ± sinθ Y Y Y Y 5 5 = ( cos θ = 6π 6π ( z x y = 5 ± iφ 5 cosθ sinθ = 8π 8π ( x + ± 5 ± iφ 5 = sin θ = π π ( x+ iy ± iy z J. F. Haison Michigan Stat Univsity 5

6 Th sphica haonics a othonoa in th sns π π '* sinθdθ d φy ( θφ, Y ' ( θφ, = δ' δ ' Radia functions Th adia functions fo a on cton ato a th soutions to th diffntia quation h ( + h Z R( ( + = ER μ μ 4πε and th dtaid soution to this quation is givn in any txt books, Pauing & Wison Th adia functions dpnd on two quantu nubs n&. n is cad th pincipa quantu nub and is a positiv intg btwn and whi is th obita angua ontu quantu nub discussd abov. Fo a givn n, is constaind to b btwn and n. Th gna fo of th adia function is in which Z R n + = L n+ ( n! {( } na n n+! (, and Z = na μ a μ 4πε = μ h and + L ( is an associatd Lagu poynoia. Not that a μ dpnds on th n+ ducd ass of th on cton ato and thus vais fo ato to ato and aong isotops of a givn ato. If w usd th ass of th cton ath than th ducd ass 4πεh of th nucus-cton syst a μ woud b th Boh adius a =. Thy a atd by aμ = a +. N J. F. Haison Michigan Stat Univsity 6

7 Th fist fw associatd Lagu poynoias a n = = L ( x = n = = L = L n = = L!( = L 4 = L 5 5 n = 4 = L 4 4!(4 6 6 = L 5 5!( 5 = L 5 6 = L 7 7 and th fist fw adia functions a shown bow. n =, K sh: n =, L sh: n =, M sh: =, s R = Z a μ ( Zaμ ( Zaμ ( =, s R = =, p R = 6 ( Zaμ ( Zaμ ( Zaμ ( =, s R = =, p R = =, d R = 9 J. F. Haison Michigan Stat Univsity 7

8 Ths functions a a noaizd to R n d = and a othogona within a paticua ' n = ' n nn R R d δ This ans that a of th s functions, a of th p functions, tc a utuay othogona but fo xap R ( s is not othogona to R ( p. Ths adia functions a pottd bow fo Z=. Th a sva fatus of th adia functions that dsv ou attntion and a iustatd in ths pots. Fist, ony th s functions a non-zo at th oigin. Scond, a givn adia function Rn ( has n nods btwn and. Thid, functions shaing th sa pincipa quantu nub n hav copaab agnituds and spatia xtnsions...5 Hydogn ato s adia function R ((au (a.8.6 Hydogn Ato s & p adia functions R (.4. R s ( R p ( (a J. F. Haison Michigan Stat Univsity 8

9 .4. Hydogn ato s, p & d adia functions R ( (au.. R s ( R p ( R d ( (a Radia Distibution Function Th adia distibution function fo an obita is dfind as th pobabiity of finding an cton on th sufac of a sph of adius with th nucus as th oigin. W can div it by fist considing th pobabiity of an cton in th stat ψ n bing found in th vou ncosd by a sph o adius. Dnoting this as P ( and noting that th sphica haonics a noaizd to w hav P d d sin d R = = ( d π π * φ θ θψ nψ n n Thn th pobabiity of finding it within a sph of adius + δ is P ( + δ so th pobabiity of finding th cton in th vou btwn th two sphs is givn by th diffnc δ + δ n n P ( + P = R ( d R ( δ and th adia distibution P ( + δ P ( function is obtaind as i = R n (. Not that ach obita has a δ δ adia distibution function (RDF that dpnd on n& but not. W pot ths fist fw RDF s bow.. J. F. Haison Michigan Stat Univsity 9

10 .6 Hydogn ato s adia distibution function.5.4 R ( (a. Hydogn ato s & p adia distibution functions.5 p s R ( (a. Hydogn ato s, p & d adia distibution functions..8 d p s R ( (a Sva ipotant fatus of th adia distibution function a iustatd in ths pots. Fist, th RDF of ψ n hav n axia. Scond, th hight of th agst axiu fo a givn n dcass as dcass. Thid, within a givn n th spatia xtnsion incass as dcass. Fouth, th spatia xtnsion incass with incasing n J. F. Haison Michigan Stat Univsity

11 Engy and Dgnacy Whi th adia wavfunctions dpnd on n& th ngy ignvau E, givn by Z E n =, and dpnds ony on n. This is a consqunc of th Couob 8πε an μ potntia. Futho sinc th tota wavfunction ψ n dpnds on n, & w ay hav sva diffnt wavfunctions with th sa ngy. An individua wavfunction ψ n dfins a stat whi an ngy E n dfins a v and diffnt stats having th sa ngy a said to b dgnat. Th nub of stats in a givn ngy v is cad th dgnacy of th v. Bcaus of th constaints on th quantu nubs &, n and th dgnacy of th v En is (+ = n. Copaison with xpint Sinc w hav th ngy vs of th atos w can copa th thotica tansition ngis with xpint. Lt s fist ook at th tansition fo th gound v to th continuu o th ionization ngy. To btt undstand th ffct of th ducd ass w wi wit a a / μ = γ wh γ = + γz γz and so En = = R wh N 8πε an n R = adius 8πε a is cad th Rydbg. Th subscipt ans that w hav usd th Boh in th dfinition. Occasionay on wi s th Rydbg dfind as a n = R μ = 8πε a so th two a at though th ducd ass γ R = R in cton vots quas γ Z V and so E n= V and th ionization ngy is n IP=-E = γ Z V. If th nuca ass was infinit, γ =, and a isotops of an nt woud hav th sa IP. W s fo th foowing tab that th is a sa but asiy asuab dpndnc of th IP on th isotop of a givn nt. Ato Z Mass (u IP(xp IP ( γ = V γ IP ( γ V Δ IP ( γ = V ΔIP ( γ H H H H R μ μ 4 H Li Li B J. F. Haison Michigan Stat Univsity

12 Th figu bow pots th diffnc btwn th xpinta IP and that cacuatd with (d and without (back th cnt of ass coction B + (IP(xp-IP(cac V H H H + H 4 H + IP(xpint-IP(γ 7 Li + 6 Li + H + IP(xpint-IP(γ= Mass (u Th o is a bit atic if w ngct th cnt of ass coction (back cuv but bcos vy ody whn this coction is incudd (d cuv and th o btwn th xpinta and cacuatd IP is ssntiay th sa fo a isotops of a givn nt. This ans that th o ust dpnd on th atoic nub Z. A itt 4 nuica xpintation suggsts that th o dpnds on Z and ths data suggst 4 4 that th coction is ΔIP( γ.5x Z V. W wi s att that th ading t in th ativistic coction to th IP is.8x concusion. ZVin asonab agnt with this 4 4 In addition to th IP w can copa th cacuatd ngy vs with xpint. Th owst tansition in th Schoding thoy is n= n= which cosponds to th ngy E E =.46895γ Z V which fo th H ato is V o 8,58.58 c -. Expintay howv instad of a sing v with this ngy w s th cosy spacd vs a within a wavnub of th Schoding pdiction as shown bow. J. F. Haison Michigan Stat Univsity

13 N- vs of th Hydogn ato ΔE(c c - P / S / P / Lab shift.5 c N= Schoding Onc again w s that th Schoding quation is akaby accuat, th o in th tansition ngy bing about c - out of 8,59 c - and ost of this o is du to ativistic ffcts which w wi not discuss. Howv th fist od ativistic coction fo th o in th ionization ngy is shown in th foowing figu and indd it aks a significant ipovnt. Th aind of th o is du to th ngct of th couping btwn th adiation fid and th ato (quantu ctodynaics. In what foows w wi consid ony th consquncs of th Schoding thoy and whi it is not pfct ths sut show that it is vy, vy good. IP(xpint-IP(cacuatd V Eo in th cacuatd Ionization ngy H H H + 4 H + H Schoding 6 Li + 7 Li + Paui (ativistic 9 B + 9 B Mass (u J. F. Haison Michigan Stat Univsity

14 Spin Whn spin is takn into account th hydognic wavfunctions a ψ nα & ψnβ and sinc th Schoding Haitonian dosn t contain spin both hav th sa ngy and th dgnacy of th v with pincip quantu nub n has bn doubd fo n to n. W wi s that spin has a pofound ffct on atos and ocus with o o ctons. Atoic Units It is usuay o convnint to xpss th ngy in atoic units ath than in th SI syst. In th atoic syst ngths a xpssd as utips of th Boh adius 4πε h a = = M which diffs fo a μ in that th cton ass has pacd th ducd ass. Th Lapacian bcos = au which suts in th a Haitonian fo th intna otion bcoing Hˆ h Z h Z = = = μ 4πε γ ˆ au au H au a aua a au a h wh γ = + accounts fo th fact that th cnt of ass is not pcisy at th N nucus. It s sast fo th hydogn ato, γ H = and appoachs as th nuca ass incass. Th Schoding quation is thn h Hˆ au ψ = Eψ o a Hˆ Ea h au ψ = E au ψ wh E au =. quas x -8 J o 7.8 h a V and is th aount of ngy cosponding to atoic unit. This ngy unit is oftn cad a Hat and psntd as E H o sipy th atoic unit of ngy. Sinc th Z aowd vaus fo th intna ngy a En = th ignvaus in Hats 8πε an γ Z γ Z a En =. Not that E = is th ionization ngy of th ato. n Sinc in what foows w wi usuay us atoic units w wi dop th au fo th ngy and vaious ngths and assu a nuci a assiv ativ to th cton so that γ =. This is an xcnt appoxiation fo ost puposs. Accodingy th on μ J. F. Haison Michigan Stat Univsity 4

15 cton Haitonian wi b wittn as Z a H ik ato is thn Hats. Hˆ Z = and th gound stat ngy fo Ra Sphica Haonics Bcaus th ngy of a hydogn ik ato is indpndnt of & w can tak ina cobinations of th dgnat ignfunctions and sti hav a vaid soution to th Schoding quation. A vy coon choic is to tak ina cobinations of th copx sphica haonics so that th angua ignfunctions a a and sti ignfunctions of ˆL but not of ˆL z. Sinc Y a aady a ths don t chang but w wi ist th again fo coptnss: s obita, = 4π p obitas, = p z = cosθ 4π + Y Y px = = sinθ cosφ 4π + Y + Y py = = sinθ sinφ i 4π d obitas, = 5 d = Y z = cos 6π + ( θ + Y Y 5 dxz = = sinθcosθcosφ 4π + Y + Y 5 dxz = = sinθcosθ sinφ i 4π d xy Y = Y i 5 5 = sin θ sinφcosφ = sin θ sin φ 4π 6π Y + Y 5 5 = = = φ 6π 6π + d sin θ x y ( sin φ cos φ sin θcos J. F. Haison Michigan Stat Univsity 5

16 Paiing an angua and a adia function suts in a vaid hydogn-ik obita. Fo xap Z ( s = R ( Y = 4π and Za 5 dxy = R (d xy ( θ, φ = sin θ sinφ 9 6π J. F. Haison Michigan Stat Univsity 6

17 Mo Hydogn-ik Wav Functions Fo convninc w tabuat th hydognic wavfunctions fo n=,,, 4, 5 & 6 and =,,,, 4, & 5. (s though h obitas ψ (, θφ, = R ( Y ( θφ, = R ( Θ ( θ ( φ n n n Φ with th sphica haonic wittn as a poduct of Θ ( θ & Φ ( φ Y ( θ ( φ ( θ, φ =Θ Φ Th functions in, θ, and φ a spaaty noaizd to unity and utuay othogona π * Φ φ Φ φ dφ = π Θ sin θ θdθ = R n d = π π * d sin θdθ ψ (,, (,, n θ φ ψn θ φ dφ = δnn δ δ vanishing, xcpt fo n= n, =, and =. =, s obitas: =, p obitas: ( θ Θ = ± ( θ Θ = ( θ Θ = 6 cos θ sin θ Th Functions Θ ( θ J. F. Haison Michigan Stat Univsity 7

18 =, d obitas: =, f obitas: = 4, g obitas: ± ( θ ( cos θ Θ = 4 5 Θ ± ( θ = sin θ cos θ ( θ Θ = ± ( θ 5 sin 4 θ 4 ( θ sin ϑ( 5cos θ ( θ 4 5 cos cos Θ = θ 4 Θ ± = 8 5 sin cos Θ ± θ = θ θ 4 Θ = 4± 4 7 sin 8 θ θ 5 ( θ sin θ( 7cos θ ( θ 9 5 cos 4 cos Θ 4 ( θ = θ θ Θ 4± ( θ = sinθ cos θ cosθ 8 Θ 4± = 8 7 sin cos Θ 4± θ = θ θ 8 Θ = 5 sin 6 4 θ J. F. Haison Michigan Stat Univsity 8

19 = 5, h obitas: Θ 5 ( θ = cos θ cos θ + cosθ 6 5 5± ( θ sin θ( cos θ 4cos θ + Θ 5 ± = 6 55 ( θ sin θ( cos θ cos θ Θ 5± = 8 77 ( θ sin θ( 9cos θ Θ 5± = ( θ 85 sin 4 cos Θ 5± 4θ = θ θ 6 Θ = 54 sin 5 θ Th Hydogn ik Radia Wav Functions Kp in ind Z = na n =, K sh: n =, L sh: n =, M sh: =, s R = Z a ( =, s R = =, p R = 6 ( =, s R = =, p R = =, d R = 9 J. F. Haison Michigan Stat Univsity 9

20 n = 4, N sh: n = 5, O sh: ( =, 4s R4 = =, 4 p R4 = + 5 = 4, d R4 = = 4, f R4 = ( =, 5s R5 = =, 5 p R5 = =, 5d R5 = = 5, f R5 = = 45, g R54 = 9 7 J. F. Haison Michigan Stat Univsity

21 n = 6, P sh: ( ( Za = 46, g R ( = ( =, 6s R6 = =, 6 p R6 = =, 6d R6 = = 6, f R6 = = 56, h R65 = J. F. Haison Michigan Stat Univsity

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