Today s topics. How did we solve the H atom problem? CMF Office Hours

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1 CMF Offc ous Wd. Nov. 4 oo-p Mo. Nov. 9 oo-p Mo. Nov. 6-3p Wd. Nov. 8 :30-3:30 p Wd. Dc. 5 oo-p F. Dc. 7 4:30-5:30 Mo. Dc. 0 oo-p Wd. Dc. 4:30-5:30 p ouly xa o Th. Dc. 3 Today s topcs Bf vw of slctd sults fo ato Mult lcto atos & Atoc spcta ow dd w solv th ato pobl? Slct a pottal fucto Coulobc Dt BCs basd o physcal asog Stup Schödg quato Assu spaato of vaabls Spaat quatos Solv quatos Apply BCs Schödg Equato fo th ato µ [ U E ] 0 E 0 µ o Rsult: Wav fuctos & Egs

2 ydog Ato vaabls : of spaato apply Now R [ ] 0 ΘΦ ΘΦ E U R R µ Ca spaat to two dfftal quatos dpdt o adus ad agls Aft sval stps w obtad: Wav fuctos fo ato haocs a sphcal wh R Sphcal aocs a gfuctos of z π π ϕ ϕ ϕ ϕ z z Θ Θ ΘΦ Agula Pat of ato pobl ylds: haocs a th sphcal wh ΘΦ Ths sult uss dfto of agula otu opato

3 ydog Ato Ths th fuctos a typcally usd as th agula pat of th ydog wav fuctos bcaus thy a al ad othogoal p x 3 4π py 3 4π p z 0 / / s cos s s ydog Ato Radal Wavfuctos / 3/! / R a0 [!] 3 a0 a0 wh th a agu fuctos a0 Valus of l ad a costad by th soluto Pobablstc Itptato of Wav fucto * dddϕ P obablty dsty Radal Dstbuto of ato s Obtal Plot [R l ] vs. Spaat to agula ad adal pats: * [ R ] d sdddϕ ϕsddϕ ow do w dt <>? 3

4 QM Egy vl Sch fo ato Egy lvl sch should pdct ato sso& absopto spcta Rydbg showd that all ato ls gv by: ν R R Rydbg' s costat c - QM Pdcts Spctu! µ 4 E 8ε 0 h 8πε0a0 K E E K f... As Chsts w wat to xtd QM to o coplx systs.g. oth atos & olculs Goal: To pdct gs & oth obsvabls Atos: Esso spcta Molculs: Stuctu vbatoal E ad otatoal E 4

5 ow wll you st up th lu Ato pobl? altoa cludg Coulobc Pottal - Nuclus Z - N Z Z o o o Objctv: to solv Schödg quato. Accptabl outcos: Soluto o good stat. st Stp: to df th pottal fucto Bo-Opph Appoxato s usd to spaat ucla ad lctoc oto Z o Z o I ths altoa w a gog ucla oto. o Schödg Equato What wll ou wavfucto look lk? ϑ ϕ ϑ ϕ Mo covtly w wll us a spl otato: Now w hav: E 5

6 Schödg Equato fo ato co t E Z Z o o Ca w spaat vaabls? What t causs th toubl? Why? What would spaatg th vaabls ply about lcto ovt? o Spaatg th vaabls: t s spaat th vaabls ayway: t s spaat th altoa too: Z o ptub. 4 πε o ptub. Z o Splfyg assuptos. Fo th t-bg lt s go th ptubato ad just th splfd altoa: Now th pobl s spaabl ad w hav a zoth od soluto: s s E E s E s Dfft Mthods fo Appoxatg th soluto:. Ptubato Thoy. W ca ty to us ths stat as th bass fo a xpaso sot of lk a Taylo ss about a pot that would clud ffcts of th ptubato.. Vaatoal Mthod.W ca vay th wavfucto usg hydogc wavfuctos as a statg pot ad s f that hlps. 3. at-fock Mthod. W ca ty to avag th oth lcto s obtal to a cloud ad us that as a pottal to st up a o-lcto pobl th pat th pocdu utl t all bcos cosstt. 6

7 Mthods fo Appoxatg soluto ca b boadly appld Ths tatts ca b galzd to systs wth o lctos: Multlcto atos wth th Bo Opph appoxato: Appoxat solutos a gally foud ucally wth coputs th bass of spcal pojct Fst w ust dvlop th cocpts udlyg thodology fo spl systs lc E Z o j >... l E o j : Atoc Esso Spcta wll b usd to tst th accuacs of th thods N: Atoc Esso Spcta lght bulbs : Na: : K: C: X: 7

8 Rvst ato pobl Ato Schödg quato wth Bo- Opph Appoxato Z o Z o o Dvlop cocpts bhd ptubato thoy I ths altoa w a gog ucla oto. E 0 Ptubato Thoy Cossts of a st of succssv coctos to a uptubd syst th altoa of th uptubd syst wh E th altoa of th ptubato 0 Ptubato Thoy Assug that th ffct of s sall th wav fucto ad th Egy lvl ca b appoxatd. E E 0 0 E 8