CHAPTER-11 The SCHRODINGER EQUATION in 3D

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1 Lt Nots PH 4/5 C 598 A. La osa INTODUCTION TO QUANTU CHANICS CHAPT- Th SCHODING QUATION in 3D Dsiption of th otion of two intatin patis. Gna as of an abita intation potntia. Cas whn th potntia pns on on th ativ position of th patis U U Dopin th Cnt of ass otion an th ativ otion Th C vaiab an ativ position vaiab Sotion b spaation of vaiabs qation of otion fo th Cnt of ass qation ovnin th ativ otion.3 Cnta Potntias U U.3A Spaation of vaiabs tho.3b Th ana qation Th Ln q..3c Th aia qation sin th Coob Potntia aia wavfntions of th bon stats < fns: "Intotion to Qant hanis" b Davi Giffiths; Chapt 4

2 CHAPT- SCHODING QUATION in 3D Dsiption of two intatin patis otion On pati otion In th as in whih a pati of ass ovs in -D an insi a potntia Vt th Shoin q. is t V t i wh = t Whn th pati ovs in th 3-D spa th qation aopts th fo t V t i wh = t h stans fo th Lapaian opato In Catsian ooinats: z In sphia ooinats: sin sin sin θ θ X Z z t

3 . Gna as of an abita intation potntia Two patis otion = z t Z = z t X Th ta intation btwn th two patis is spifi thoh th potntia V t an th osponin Shoin q. is i V t t 3 t wh = In th pssion abov th foowin notation is bin s z ; ; t z. Cas whn th potntia pns on on th ativ position of th patis H w onsi that th potntia V t pns on on th ativ position t = t 4 V U Dopin th Cnt of ass otion an th ativ otion Fo this patia as it an b shown that th otion of th two patis an b op into th otion of th nt of ass an th otion ativ to ah oth. Th C an th ativ-position vaiabs Sh opin an b ahiv thoh th foowin han of vaiabs: wh 3

4 4 5 t = t Noti b ain z an z th finitions in 5 an b -wittn o piit as z z z z z 6 t = t wh w a sin o a want to p to pssion whih ivs th fo in whih q. 3 is tansfo pon th appiation of th han of vaiabs fin in 5. t = t ipis

5 5 z z z z z Sin z ain th ast th sts ivs z z wh w hav s th notation z z an z z V onvnint th pvios sts is pss as 7 Siia

6 6 Anaoos sts fo an z a to 8 pain 7 an 8 in 3 on obtains t U t i In ts th ass 9 on obtains th qation t U t i wh = t Sotion b spaation of vaiabs W wi spiaiz in th as wh th potntia U os not pn piit on th ti t. Fo sh a as w wi b ooin fo sotions to q. in th fo t t C i F / t t i i F C / / [ [ pain in F U F F F C

7 Diviin b F C F U F This pssion iniats that th ts on th iht athoh intinsia pnin on th vaiabs an th a p to a onstant va. Stiv w hoos C an F U 3 F qation of otion fo th Cnt of ass Th fist pssion as to 4 C whih ait sotions of th fo i / n C 5 wh n is an abita onstant nit vto. That ans th a an sotions of this tp on fo ah st nit vto n. In ts of th wavvto is wittn as = i. C C n th pvios pssion C 5 In shot if a sin vto C w s in th spaation of vaiabs ivn in thn th otion of th C wo b sib b a pan-wav that popaats in th ition of C / t i[ - / t] C C. i C 6 7

8 Aoin th C is i to b anwh in th spa bt has a v finit ointation of its ont that ivn b. C If a o pis oaization of th C w si thn w an C st a bnh of wavvtos C n within a an an fin a wavpat i [ - / t] C C t. 7 C That wa th C wi b o oaiz bas t is o spatia oaiz bt at th sa ti th wi b a osponin ntaint in its ont. It is woth ntionin that w hav not bn ooin fo a sotion t as w wo hav bn oin in a assia hanis appoah; ath w hav fon an apit pobabiit. qation ovnin th ativ otion Ba to q. 3. This as to F U F F 8 In th nt stion w wi spiaiz fo th as fo nta potntias; that is whn th potntia pns on on th anit of th patis spaation: U U.3 Cnta Potntias U U F F U F 9 This is an qation fo F wh th n paat is sti to b tin. Whn woin with nta potntias it is o onvnint to s sphia ooinats to sov 9. At th binnin of this Chapt th Lapaian opato was ivn in sphia ooinats. Usin that pssion in 9 on obtains 8

9 9 F sin F F U.3A Spaation of vaiabs tho W wi oo fo sotions of th fo F This as to sin U Diviin ah t b sin U U + sin

10 U + + sin Th fist two ts pn on on whi th thi t on on th ana vaiabs. This ans that th aia pat an th ana pat a ah onstant. sin - U aanin ts on obtains sin Ana q. an U aia q. 3 In sa w a ooin fo sotions to q. in th fo F.3B Th ana qation To sov th ana qation sin w wi oo aain fo sotions of th fo of spaat vaiabs 4

11 + sin Diviin b an tipin b sin [ + sin ] + = This ans that ah of th two t st b onstant = [ + sin ] = 5 = - 6 Th va of th onstant onsiations. In fo th son q. abov on obtains + = is tin b phsis-bas wh th patia ivativ sbo has bn opp sin th fntion pns on on on vaiab whih has sotions of th fo = i. Bt noti that th wavfntion sho ta th sa va whnv is innt b ; that is = phsis-bas qint This ipis i = i o i = This onition an b satisfi on if is an int nb. Ths th pnn of th wavfntion on th vaiab is ivn b = i ; = - - 7

12 wh th fato / has bn into to ns th othonoait onition of ths fntions in th an * 8 W wi s bow so stitions on th aow an of vas fo s pssion 4. Z z X Th oth ana qation 5 has th fo sin 9 wh it has bn phasiz that th sotion pns on th paats an. Th tas is to fin a sotion ov th an. It is onvnint into th nw vaiab w = os an th nw fntion G w = fo. It an b shown that th q. fo G is G w G 3 w G - w w - w

13 Cas = W wi s at that th fntions G osponin to an b bit pon G. Th att satisfis th q. G G w w G Ln q. 3 w w Pow sis sotion G w = w Fo 3 on obtains G w w G w w w whih ivs 3 G w w G w + w = Aso G w w w 33 G w w w - G - w w w - w = Usin - = = w - w - w w wh w hav pa fo aain to avoi witin too an vaiabs 3

14 = [ w 34 pain 33 an 34 in 3 on obtains [ This qis ] w + ] w = w [ [ ] = sion ation 35 Ths th sotion 3 an b wittn o piit as G w = 3 4 w w w w wh an a abita onstants. pssion 3 povis two inpnnt sotions. Noti ah sbsqnt offiint pns on th va of th pvios on fo ap if on offiint w qa to zo thn a th oth sbsqnt offiints wi b n as w. Divn at w =. Fo a vas of this sis vais as / / a bhavio whih is siia to th ivnt sis /. This iniats that 3 ivs fo w= ths onstittin an naptab sotion sin th wavfntion has to hav a finit va at =9 o. Ponoia sotions. It is sti possib to obtain a satisfato sotion ot of 3 if w hosn of th fo = + with bin an int va thn th offiint sis 3 wo 4

15 bo a ponoia of with th osponin o onvnint hosn qa to zo pnin on whth is o o vn sptiv. Sa: Th Ln qation - G w w G Ln q. w G w aits phsia aptab sotions povi that has th fo of = + with bin an int. Fo a ivn th stin sotions is a ponoia of. G w = w P w wh Ln Ponoia. Sin an abita tipiativ onstant o aopan to ths sotions th Ln ponoias a fin sh that P = Fo pssion 36 on obtains = : P w 37 = : P w w = : P w 3w = 3: P w w w 3 3 = 4: P w w w w w

16 Sin P is a ponoia of P -w = - P w 38 Withot poof w stat that th Ln ponoias satisf P w P w w 39 Th na as: W a ooin fo sotions to q. 9 whih is inpnnt of th sin of. G Fo that ppos t s fin th Assoiat Ln fntions G - w w G w - w w P - / I I I I w w I I P w w II 4 That is th ponoias P I I w a obtain b tain ivativs of th aa nown Ln ponoias P w ivn in 37. It tns ot ths Assoiat Ln fntions satisf q. 9 PII II - w P w II P w w - w 4 Noti that sin w intia qa to zo fo P is a ponoia of I w I I wi b + assoiat Ln ponoias P I wi b. Aoin fo a ivn th = Usin th Ln ponoias ivn in 37 on obtains 6

17 = = : P w P w 43 = = : = : P P P w w w w P = = : P 3 w P w w / / w - w w - w / = : P / w - w P w - w 3w w = : P - / w w P - w w 3 w II Sin P w wi / I is a ponoia of - I I an - II is an vn fntion thn P P w I - w - II II w 44 I - Withot poof w stat that th assoiat Ln ponoias satisf I I I I II! P w P w w 45 II! This pssion wi b sf fo pop noaizin th wavfntion. Sa: Th ana qation aits sotions of th fo: sin 7

18 I = I I I P w P os = = 3 i ; = - - A pop onstant in font of is st so that th sotions a noaiz. Th stant fntions a a th sphia haonis. Sphia Haonis /! i P os! fo 46 * fo Th fato - is hosn fo onvnin oth athos notation iff b a fato of i. Th sphia haonis onstitt an othonoa st of fntions Usin * ana spa sin sin * 47 Th foowin wbsit is v hpf fo visaizin th sphia haoni fntions: = = 8

19 4 / = = = = Absot va of a pat Absot va oo b op phas = = 3 4 / os = = Absot va Absot va Th two iffnt oos iniat th iffnt sin tan b th wavfntion = = ± 3 sin 8 / i 9

20 = = = = Absot va of a pat 3 8 / sin os Absot va oo b op phas = = - Absot va of op pat / 3 sin sin / 3os 5 8 / sin os i / 5 sin os 3 i

21 = = = = = = Absot va of a pat iinat b iht. = = = = = = Absot va oo b op phas Th sphia haoni fntions ivn in 46 onstitt a opt st of sotion of th ana qation. W a now ft with th tas of sovin th aia qation 3..3B Th aia qation W hav fon that in q. 3 that th aia qation st satisf U

22 wh tas int vas = this onition os fo th sotions qi fo th ana oponnt of th wavfntion an th vas fo n to b tin. Fo th as of a Coob fi U Z wh Z is th atoi nb nb of 4 w hav potons in th ns an is th nta ha 48 Usin nit-ss vaiabs Whn onsiin th as of a Coob fi it is onvnint to s 3 49 as th nits of ass nth an ti sptiv. Noti that th nit of n is. Fo o ppos insta of w wi s th ass insta. Ths sin has nits of istan an has nits of n th qation abov an b pss in ts of nit-ss paats an insta of th vaiabs an wh. 5

23 3 In sin in aition = t. on obtains 5 Lt = 5 Th ppos of this han of vaiab is to obtain an qation th sbs th on insiona as as w wi vif bow Ain th ast two pssions This q. an b pss as

24 V 53 ff wh an fftiv potntia has bn fin as V ff V ff = Noti q. 53 sbs th as fo a on insiona pob. Howv It has sinifian on fo > an st b sppnt b bona onitions at =. W wi qi that th aia fntion th oiin. Sin Noti in th ion w st qi that ain finit at 54 th qation 53 bos Fo > th bhavio of in this ion wo b osiato. i 4

25 5 If o intst is to fin bon stats w an ns that th wavfntion vanishs whn if w qi that < Looin fo bon stats < Bon stats wi b obtain with nativ vas fo. Lt s fin wh is a paat to b hosn onvnint at. =v v v v v v v v - - Choosin v 4 v - Lt v 4 v - Sa

26 6 wh 4 Z = = with an ivs =v an ivs v 4 v - 55 with th onition v Th fntion v an th vas of th paat assoiat to th n n to b tin. Lt s fi ot th bhavio of v na th oiin. Ass s v =... s s o 56 Th pow s is nnown i.. w o not now t has fast v tns to zo whn. Th va of s is tin b pain th pansion 56 in 55 an anazin th ts of owst pow that is s. In fft sh po ivs

27 s s o o s s s s s s s s s Fo a ivn va of this pssion abov iposs th foowin onition on s s s onition on s 57 Th possib hois fo s a s = + an s = -. Th son option is isa on th basis that sh stion wo not satisf th onition v. Ths s = + v Lt s fi ot th bhavio of v whn. In this ion q. 55 tas th fo v - v 4 58 whih has sotions of th fo v 59 F sotion of q 55 Th sts in 58 an 59 sst ooin fo sotions of th fo v 6 wh an 7

28 8 v 4 v Ths 4 - ] [ v v Th aition of ths ast two pssions sho b qa to zo ] [ ] [ - ] [ ] [ - 6 Aoin to 6 w pan th fntion in a pow sis wh 6

29 ain - ain - ain - o pain th pssion on th iht si in 6 an nain ba to w obtain [ ] [ - ] [ - ] [ ] W obtain th foowin n foa - 63 Fo a of th fntion. This onvn is siia to th pansion! whih is not aptab in this as sin th wavfntion sho tn to zo in th ion. On wa to obtain a sf sotion is to fin th onitions n whih th sis 6 ts tnat into a ponoia. 9

30 pssion 63 iniats that that is possib if th t that tins th n of th sst w qa to an int nb. In Fo a ivn b stin = n with n 64 bos a ponoia n of o n Fo a ivn n th possib vas fo a = 3 n-; 65 That is iffnt sotions wi b assoiat to th sa n v. Fo 55. An int va fo = n ipis that th aow vas fo th n a ist an n n Z Fo 5 wh. Ths 4 n Z - 66 n 4 n wh n = 3 In th sa sptosop notation th n vs a spifi b two sbos: Th fist ivs th va of th pinipa qant nb n. Th son is a o tt s p that iniats th va of th obita ana ont... sptiv. 3

31 V s s 3p p s = = = =3 =4 aia wavfntions of th bon stats < = = = v wh n n n n/ n n wh n an Z ; 4 Th pssion fo is fth pss as Z Z 4 / a wh a.59 n n That is 4 3

32 Z / a o n n 67 n n Z wh / a an n 4 a n with an - n W hav oitt th offiint in font of th ponntia t in n sin it wi b absob b th offiint o fo th ponoia n. Cas n= Aoin to 65 th on possib va fo th obita ana ont is =. Aso fo 67 - n - Fo = th n foa ivs =. Hn = fo an >. Ths = o. pssion 67 ivs / Z 4 ; a 68 Z wh th st has bn pss in ts of th Boh s ais a

33 Th onition of noaization qis. Usin Z / a 68 on obtains ; a / Z 3 ; 3 4 Z / a 4 ; an fina Z / 3/. Ths 68 bos a 3 / Z Z / a 7 a Th aia istibtion fntion D n D 7 ivs th pobabiit p nit nth that th ton is to b fon at a istan fo th ns. a This aia sotion 7 is opnt with th sphia haoni osponin to = an = ivn in pssion 46. Ths th sotion to th Shoin q. 9 is / 4 Z / / ivn b 3 Z a F. a / 4 F Z a 3 / Z / [ ] 7 a 33

34 Cas n= Aoin to 65 th on possib va fo th obita ana ont is =. 34

The local orthonormal basis set (r,θ,φ) is related to the Cartesian system by:

The local orthonormal basis set (r,θ,φ) is related to the Cartesian system by: TIS in Sica Cooinats As not in t ast ct, an of t otntias tat w wi a wit a cnta otntias, aning tat t a jst fnctions of t istanc btwn a atic an so oint of oigin. In tis cas tn, (,, z as a t Coob otntia an