Today s topic 2 = Setting up the Hydrogen Atom problem. Schematic of Hydrogen Atom

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Marylou Nicholson
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1 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

2 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 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

4 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

5 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 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 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

7 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

8 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

9 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

Chapter 11 Solutions ( ) 1. The wavelength of the peak is. 2. The temperature is found with. 3. The power is. 4. a) The power is

Chapter 11 Solutions ( ) 1. The wavelength of the peak is. 2. The temperature is found with. 3. The power is. 4. a) The power is Chapt Solutios. Th wavlgth of th pak is pic 3.898 K T 3.898 K 373K 885 This cospods to ifad adiatio.. Th tpatu is foud with 3.898 K pic T 3 9.898 K 50 T T 5773K 3. Th pow is 4 4 ( 0 ) P σ A T T ( ) ( )