Today s topic Sttig up th Hydog Ato pobl Hydog ato pobl & Agula Motu Objctiv: to solv Schödig quatio. st Stp: to dfi th pottial fuctio Schatic of Hydog Ato Coulob s aw - Z 4ε 4ε fo H ato Nuclus Z What is difft about th Quatu chaical tatt? What is th pottial fuctio of th hydog ato? Coulobic itactio is ot kow ad caot b spcifid
Sttig up th Schödig quatio s ct-of-ass coodiats: lc lc uc lc uc Wit Schödig quatio i sphical coodiats: Schödig quatio icludig Coulobic Pottial [ ] 4 ε o Hydog Ato ε o 4 si si si ϕ ϑ Wh How will w appoach solvig this pobl? Solvig th Hydog ato pobl Assu spaatio of vaiabls: aag Schödig quatio ito 3 idpdt difftial quatios Dfi bouday coditios basd o physical asoig ϕ ϑ ϕ ϑ Φ Θ
3 Bouday coditios fo H ato ϕ ϕ Φ Φ Costaits o Θ ais i solvig th quatio ad quiig that th wav fuctio b wll bhavd i.. quadatically itgabl. S vi Sc. 5.3 fo dtails. Hydog Ato: Spaatig Vaiabls [ ] si si si { { : by Schodig q'. Multiply [ ] si si si call fo fo agula otu opato: Substitut ito Schödig quatio: Hydog Ato vaiabls : of spaatio apply Now [ ] [ ] : by though ΘΦ ΘΦ ΘΦ ΘΦ ΘΦ ΘΦ Divid Ca spaat ito two difftial quatios dpdt o adius ad agls
Agula Pat of H ato pobl is th sa as igid oto call th solutio fo Φ i th igid oto { wh a th sphical haoics Φ i ± ±... Solv th difftial quatio Applid Bouday coditios Applid oalizatio coditio igid oto Th solutio fo th difftial quatio dpdt o ust b solvd usig a pow sis. Th solutio is th gd Polyoials.! Θ! call that liits o cos ± / P cos Th wav fuctios fo th igid oto a call Sphical Haoics /! P cos 4! Wh P cos a th i gd polyoial s Th solutio ivolvs placig costaits o th poptis of th wav fuctio: quadatically itgabl S vi Chpt. 5 fo dtails. 4
Sphical Haoics a igfuctios of Ĥ ad z call that K ad H K I { H I { { total agula otu igid oto Thus th sphical haoics a also igfuctios of: i i i i z z z pojctio of ag. o. is Th valus of ad l a latd sic z- pojctio ust b lss tha total agula otu sic Total Agula Motu ad th z- pojctio of agula otu ca b asud siultaously [ ] [ ] [ x y z ] z y x 5
6 Oth copots of agula otu ca NOT b asud siultaously ] [ ] [ ] [ y i x z x i z y z i y x It ca b show that adial Wavfuctio fo H ato Not: idpdt of Hydog Ato... 8 8 4 thod to giv : pow sis quatio ust b solvd with a This h a wh a h ε ε ε / 3/ / 3!] [! a a a a agu fuctios th wh a Hydog Ato that is : ad othogoal a poply oalizd Th * si d d d d d d δ δ δ si
3 Hydog Ato Th fist fw agu Fuctios a Boh adius x x! a 3 3 x 3! 3! 3 x 33 3a 3a Th fist fw Hydog Atoic Wav Fs Zatoic ub of th uc lus; σ Z / a ; a Boh adius 3/ Z σ a 3 / Z σ σ / 3 a 3 / Z σ σ / cos 3 a ± 3 / Z σ σ / si ± i 64 a 3 / Z 7 8 σ / 3 3 σ σ 8 3 a Hydog Ato Agula otu dpds tily o gy dpds oly o th picipl quatu ub Th z copot of agula otu is dtid tily by z ħ Hydog Ato Th pobability that lcto lis btw ad d is [ l ] d Th adial fuctio has -l- ods <> <> ad <K> -<> gy lvls hav l dgacy i th absc of a agtic fild 7
Hydog Ato Th sphical haoics fo a coplx ad thus caot b pstd pictoially. Howv w ca tak lia cobiatios of sphical haoics with th sa gy to giv w igfuctios which also hav th sa gy. Hydog Ato Ths th fuctios a typically usd as th agula pat of th Hydog wav fuctios bcaus thy a al ad othogoal p x 3 4 py i 3 4 / / si cos si si p z 5 d z 6 Hydog Ato / 3cos d xz 5 4 d yz i 5 4 / / d x y 5 6 d xy i 5 6 si cos cos / si cos si / si cos si si Hliu Ato Th hliu has two lctos aoud th uclus so th S is giv by: 4 ε 4 ε 4 ε 8
Hliu Ato Bcaus th uclus is so assiv w ca gad it as fixd Thus: 4 ε 4 ε This quatio caot b solvd xactly bcaus of th itlctoic pulsio t. Nothlss w ca us a appoxiat thod to obtai a solutio to ay pcisio dsid. 9