BORH S DERIVATION OF BALMER-RYDBERG FORMULA THROUGH QUANTUM MECHANICS

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1 BORH S DERIVATION OF BALMER-RYDBERG FORMULA THROUGH QUANTUM MECHANICS Musa D. Abdullahi Umaru Musa Yar adua Univrsity, P.M.B. 18 Katsina, Katsina Stat, Nigria musadab@utlk.cm Abstract Accrding t classical lctrdynamics th Ruthrfrd s nuclar mdl f th hydrgn atm shuld b unstabl. But th hydrgn atm is vry stabl with missin f radiatin at discrt frquncis. Th Balr-Rydbrg frmula givs th wav numbr f th lins f th spctrum f th hydrgn atm in trms f th Rydbrg cnstant R and quantum numbrs n and q. Nils Bhr brilliantly stabilisd th Ruthrfrd s atm by invking quantum chanics and making tw ad-hc assumptins t driv th Balr- Rydbrg frmula. Th spctral lins rang frm th far infra-rd t ultravilt rgins. A rcapitulatin f Bhr s drivatin is givn in this papr. Kywrds: Angular mntum, hydrgn spctrum, rbit, quantizatin, radiatin, wavlngth. 1. Intrductin 1.1 Ruthrfrd s Nuclar Mdl f th Hydrgn Atm Lrd Ernst Ruthrfrd (1911) [1] prpsd a nuclar thry f th atm cnsisting f a havy psitivly chargd cntral nuclus arund which a clud f ngativly chargd lctrns rvlv in circular rbits. Th hydrgn atm is th simplst, cnsisting f n lctrn f charg and mass m rvlving in a circular rbit arund a much havir cntral nuclus f charg +. This mdl, cncivd n th basis f xprintal rsults, has sufficd sinc, althugh with s difficultis rgarding its stability and mittd radiatin. Accrding t classical lctrdynamics [] th lctrn f th Ruthrfrd s mdl, in bing acclratd twards th nuclus f th atm, by th cntriptal frc, shuld: - 1 -

2 (i) (ii) mit radiatin vr a cntinuus rang f frquncis with pwr prprtinal t th squar f its acclratin and ls ptntial nrgy and gain kintic nrgy as it spirals int th nuclus, lading t th cllaps f th atm. Th scnd prdictin is cntradictd by bsrvatin as atms ar th mst stabl bjcts knwn in natur. Th first ffct is cntradictd by xprints as a dtaild study f th radiatin frm hydrgn gas, undrtakn by Jhann Balr, as arly as 1885 [3, ], shwd that th mittd radiatin had discrt frquncis. Th spctral lins f radiatin frm th hydrgn atm satisfy th Balr-Rydbrg frmula; 1 æ 1 1 ö n nq Rç - (1) lnq è n q ø whr lnq is th wavlngth, nnq th wav numbr, n and q ar intgrs gratr than 0 with q > n and R is th Rydbrg cnstant. This imprtant frmula was first btaind by Jhann Balr (1885), as a spcial cas fr n, and thn gnralisd by Jhanns Rydbrg (1888). Fr n 1 and (q - ) w hav th Lyman sris in th far ultra-vilt rgin; fr n and (q 3 - ) thr is th Balr ( visibl lin) sris and whr n 3 and (q - ) w gt th Paschn sris in th nar infra-rd rgin. Othr sris fr n > ar in th far infra-rd rgins. R was fund, by asurnts, t b 1.097x10 7 pr tr [5]. Equatin (1) givs th spctral sris limit (n ) as nn R/n. 1. Bhr s tw pstulats Nils Bhr ( ), in a suprb display f riginal thught, rscud th hydrgn atm frm radiating and cllapsing. H drivd th Balr- Rydbrg frmula, fr th spctral lins f radiatin frm th hydrgn atm, by invking th quantum thry and making tw pstulats [6, 7]. Bhr pstulats ar as fllws: i. th lctrn, in th Ruthrfrd s nuclar mdl f th hydrgn atm, can rvlv, withut radiatin, rund th nuclus in allwd, quantum r stabl rbits fr which th angular mntum Ln is quantizd, such that: Ln () p - -

3 whr n, th quantum numbr, is an intgr gratr than zr, and h is th Plank cnstant. ii. Th scnd f Bhr s pstulat is that an xcitd lctrn translats frm a stabl rbit f radius rq, crrspnding t quantum numbr q and ttal nrgy (kintic and ptntial) Eq, t an innr rbit f radius rn, crrspnding t quantum numbr n and ttal nrgy En. Th lctrn lss ptntial nrgy and gains kintic nrgy and, in th prcss, it mits radiatin f frquncy fnq, in accrdanc with d Brgili s hypthsis, such that: E - E hf (3) q n nq whr n is a numbr gratr than zr but lss than q.. Drivatin f th Balr-Rydbrg Frmula Lt us apply th tw pstulats f Bhr t th hydrgn atm whs lctrn f mass m and charg at a pint P rvlvs with vlcity vn abut a statinary nuclus f mass M and charg + in a circular rbit f radius rn, as shwn in Figur1. Th angular mntum f th lctrn, in th nth rbit, is: Figur 1 An lctrn f charg and mass m at a pint P rvlving with spd vn, thrugh an angl ψ, in a circl f radius rn undr th attractin f a havy nuclus f mass M and charg + at th cntral pint O

4 Ln mvnrn p Equating th cntral frcs n th lctrn, w gt: mvn rn p rn Equatins () and (5) giv th spd and radius f rvlutin as: vn rn (7) p With n 1, w gt th Bhr radius r m. Th ttal nrgy, in th nth quantum stat, is btaind as: 1 En mvn - (8) p rn Substitut fr vn and rn frm quatins (6) and (7) givs: En - (9) Th ttal nrgy in th qth quantum stat is: Eq - (10) qh Equatin (3) thn bcs: æ 1 1 ö Eq - En (11) ç - hfnq h èn q ø fnq 1 æ 1 1 ö (1) 3 ç - c lnq 8c h èn q ø Equatin (1) is th Balr-Rydbrg frmula (with n > 0 < q) fr th spctral lins f radiatin frm th hydrgn atm. Th Rydbrg cnstant R, in quatin (1), is givn by: R (13) 3 8c h Substituting th valus f th physical cnstants in quatin (13), R is fund as pr tr, in agrnt with bsrvatin [7]. - - () (5) (6)

5 3. Cncluding Rmarks Th stabilizatin f Ruthrfrd s nuclar mdl f th hydrgn atm, by Nils Bhr, was rcgnisd as a brilliant achivnt f th human intllct. It gav an additinal imptus t th dvlpnt f quantum chanics. Hwvr, it has s drawbacks as pintd ut blw: (a) Th radius f circular rvlutin f an lctrn, bing prprtinal t n (quatin 7), is nt quantizd. (b) Th transitin frm n rbit t anthr, n n quantum jump, in zr ti, as a ncssary cnditin fr radiatin f nrgy. (c) Failur t rlat th frquncy f mittd radiatin t th frquncy f rvlutin f th lctrn, rund th psitivly chargd nuclus. Equatin (1) is an xampl f Bchmann s Crrspndnc Thry, whrby th xpctd, dsird r crrct rsult is mathmatically btaind, but n th basis f wrng undrlying principls. In thr th mathmatics is crrct but th physics is wrng and thr can nly b n crrct physical xplanatin. Rfrncs [1] E. Ruthrfrd; Th Scattring f α and β Particls by Mattr and th Structur f th Atm, Phil. Mag., 1 (1911), 669. [] D.J. Griffith; Intrductin t Elctrdynamics, Prntic-Hall Inc., Englwd Cliff, Nw Jrsy (1981), p 380. [3] [] Francis Bittr; Mathmatical Aspcts f Physics, Anchr Bks, Dublday & C. Inc., Nw Yrk (1963), pp [5] N. Bhr; On th Cnstitutin f Atms and Mlculs, Phil. Mag,, Sris 6, Vl. 6 (1913), pp [6] [7] L. D Brgli; Ann. Phys. (195), 3,. [8] Ptr. Bckmann; Einstin Plus Tw, Th Glm Prss, Buldr C. (1987) - 5 -