# 4.1 Schrödinger Equation in Spherical Coordinates

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1 Phs 34 Quu Mehs D 9 9 Mo./ Wed./ Thus /3 F./4 Mo., /7 Tues. / Wed., /9 F., / Shodge Sphe: Sepo & gu (Q9.) Shodge Sphe: gu & d(q9.) Copuo: Sphe Shodge s 4. Hdoge o (Q9.) 4.3 gu Moeu Sp ½ & Mge Feds (Q5.5,6.-.,.5) ddo o gu Moe D.M D.W Week D.F D 9.M Week 9 D 9.W D 9.F qupe Gh s e Pou o ose wh wh pues I hve Wheods d pes Sphe Shodge hdou Copue ued up wh hp:// Chek des ouees: Pog ssge o e e s pog uo D.M Mod / Ghs Shödge Sphe: Sepo & gu (Q9.) 4. Shödge quo Sphe Coodes Gog o 3-D Ces s e s ou d epe o e. he h hvg jus oe opoe o oeu, hee e hee; og wh h, he h jus kg he devve wh espe o, ou ke wh espe o d oo. Css, p p p p V V So, sg hs o Shodge s equo,, O ouse, hs oo o devves shoud ook s,,,,, V,,,,, d hus

2 Phs 34 Quu Mehs D 9 V og wh ovg ou equo o 3-D; ovous he wves ow ve 3-D.. Coepu: We he oeu opeo 3 desos es o u veos h, h, d h. pe: 3-D Ie Sque We. Pep o ss: ed poe 4.; we w do hs poe ss.. Coepu: Se up he poe. Whee do ou s?. Mh: How woud he souo hge he o s o u? S he pe s oed o: < < d < < d < <. Fo epe, hk o he 3-D e sque we, o wdh, deph, d hegh. Woud oe e u supsed ws soved e? Hek, e s pug d see h woks d see wh he eeg epesso s How oud we epese suh wveuo? Wh oo es pg he oe o he ouh deso, ed posve, gee egve, soehg ke hs: hp:// Noo: g, he sque o hs hg ees o he po o dg he pe soewhee, u ow soewhee s houghou 3-D voue he h jus og -D e. So sug ove posses o ge po o es dog voue eg.

3 Phs 34 Quu Mehs D 9 3 spe spe ddd ddd dvo.. O ouse, ou se he wveuo s sp ousde he o, so h eeve shks he s o ego o d d d ddd Jus oug o oe o hese hee egs s, 4 os d d S o he ohe wo egs, so Vo So, e Pehps hs ooks o Ss Mehs: eg pe oo s pe 3-D e sque we. 4.. Sepo o Ves Now, hs se I jus spped, whou sg hg ou, h ou souo woud ke e sepe;, hs souo hs he o o whee eh o he souo s sp he souo o -D poe,, e. hough he e ked h Ohe Coode Sses. Now, s I sue ou e, Ces oodes e hd ou poe hs egu se (o k heeo), whe d oodes e hd ou hve

4 Phs 34 Quu Mehs D 9 d se (ke he ee ed o hged od o he ge ed o e ue), d sphe oodes he hd ou hve sphe se (ke he ee ed o po hge.) Now, we e gog o ge pe eesed e poes. Th s, poe eeges h deped o how p wo ojes e,, u o he deo o he sepo. Moe he, V( ). So hs poe hs sphe se, d we w o use sphe oodes. Noe h whehe we e ookg o ege ove spe o deee uo s ves hough spe, s he se s dee oks: es sep eh ohogo oode deo; he d, d, d d 3 : dvo = d d d 3 d ˆ ˆ ˆ 3 3 (whee he h deoes u veo he h opeo so) Ces: d d d Sphe: d d d dvo = ddd ˆ ˆ ˆ o gde d dvege ˆ ˆ ˆ ˆ ˆ ˆ dvo = d d d. Coepu: Wh he s o ego whe ou ege ove spe sphe oodes? Wh? ˆ ˆ Doue hek; s hee soehg ou sug o u? ˆ o gde Fo dvegee we hve o eee e wh we e dvegee he s pe, d s e oe oped Sphe: 4

5 Phs 34 Quu Mehs D 9 5 Vo sdes Vo (oe: hs oud e deved dog hge o ves o Ces o sphe, u s p d, whe poss essug, o e egheg.) The he Lp s s he dvegee o he gde, u s e oe oped h Ces. The s sgh owd o do hs o he gde. So, he Shodge quo Sphe (d wh e poe) ooks ke V V. Mh: F sg seps he devo o equos 4.6 d 4.7. quesos? Now, jus ke we ossed ou ges d hoped h he e d poso depedee o he wveuo oud e seped o sepe os, d ke he,, d depedee oud e seped o he 3-D e poe, we shoud ook o he poss o sepg, d. Fs o, we ous o sepg o, so we guess o,,,, Puggg d eg he dee os s os o he ohe devves, V Dvdg, d eg up e, V

6 Phs 34 Quu Mehs D 9 Lke hs, os he gh depeds o u o o o ; os he e depeds os o d u o o. Bu e hese wo sdes e equ o eh ohe; s whoe, he e deped o o he ves; do o he gh. So, he us equ oo os. Wh soe oesgh, Ghs wes h os he o (+). V. Coepu: Wh ves he poe eeg he Shödge equo deped o ( ohe wods, V eq. 4. s gee uo o wh)? Is equo 4. vd V s uo o e? Wh? Is equo 4.4 vd V s uo o e? Wh? I he devo o equos 4.6 d 4.7 wh s V uo o? Wh e uo o? 4.. The gu quo Ghs goes e he gu equo s. Jus o g uo up o he ueo, Hek, ow h we ve go se o sep, ou po eoge h he d depedee s sepe. Puggg hs d dvdg, we hve depedee B he se kd o gue we jus de whe sepg o he depedee, he e o hs, os depeds o d he gh os depeds o, u he wo sdes equ eh ohe so he us, s whoe, deped o ehe, h s, equ os. Coveo s o h (he eso o he sque w eoe evde sho). 6

7 Phs 34 Quu Mehs D 9 So, oug o he gh-hd sde s, hs dee equo ooks pe d. o Se we do hve eso o eeve h he wveuo s sp pu ge, guesg oes o es o e he es, so we go wh epoes. e does he jo, d s ed oed! I we eque h he wveuo e ge vued spe, he hs eeds o hve he pope h ou eee o s week s hoewok h hs s es ssed,,,... Depedee Now, he equo s o so spe. Ghs does eve peed o ed us hough devo o h souo; sed he es d ges us wh. I k up es oe sep o ove e. The equo s eh oo whee hee s d, e s up d dvde. Now, e h d dos So we we (e wh e) os os Fo h e, see hose e squeds? We ewe he s oes os hs po, s evde h we shoud ephse he he-depede uo s oe o he depede uo. d o he ske o o kg hs ook wose, e s dee os os os os 7

8 Phs 34 Quu Mehs D 9 os (Ghs uses, u I do w ouso wh hs eg, oh, s, oode.) So he dee equo h eeds sovg s Pug ogehe equos 4.7 d 4., we hve P! / No, h ws se-evde, u ookg he o o he dee equo, s e ges pos o pus.. Coepu: Ths e Gh s sp gves ou he souo o dee equo (4.6 soves 4.5). Ls soe o he popees o hese souos. Le s ook (see Pho pog P.p). Coepu: Wh e he posse vues o? Wh? Mgude o gge h euse... So s o + >.

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