Chapter 7. A Quantum Mechanical Model for the Vibration and Rotation of Molecules

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Charles Singleton
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1 Chaptr 7. A Quantu Mchanica Mo for th Vibration an Rotation of Mocus Haronic osciator: Hook s aw: F k is ispacnt Haronic potntia: V F k k is forc constant: V k curvatur of V at quiibriu Nwton s quation: t F k iff. qn Soutions: = Asint position k vibrationa frqunc

2 Vrif: hs k Asin t A cost A sin t t t Asin t k rhs Montu: p t Acost Enrg: E p V Acost A cost A ka k Asin t sin t Whn t = = Whn t = p = an p = p a = A an = a = A Enrg consrvation is aintain b osciation btwn kintic an potntia nrgis.

3 3 Schröingr quation for haronic osciator: E k Enrg is quantiz: E... whr th vibrationa frqunc k Rstriction of otion as to uncrtaint in an p an quantization of nrg. Wavfunctions: H N whr 4 k Noraization factor

4 4! N H is th Hrit ponoia H H 4 H... H H H rcursion ration t s vrif for th groun stat N 4 k N k N k N k N hs Not that

5 k 4 It is not ifficut to prov that th first tr is zro an th scon tr as. rhs N So rhs hs it is a soution. On can aso show noraization: N 4 whr w hav takn avantag of th vn str of th Gaussian a a an us th intgra forua a 4a For highr stats th wavfunctions ar ssntia th prouct of a Gaussian an a ponoia: 5

6 Th Gaussian is awas an vn function but th ponoia can ithr b vn f f or o f f. As a rsut th wavfunctions ar ithr vn or o pning on. Proprtis of th osciator i. Zro point nrg E quantu ffct Evn at K thr is sti vibration. ii. Orthonorait: * iii. Avrag ispacnt * 6

7 No attr if is vn or o is vn an thus th intgrant is o. Intgration of an o function in is awas zro. iv. Avrag ontu * i Bcaus th rivativ of an vn function is an o function an vic vrsa. v. Nos Th stat has nos u to Hrit ponoia vi. Enrg sparation E E E qua sparation. vii. Tunning Wavfunction pntrats into cassica forbin rgions. A quantu phnonon. Eap: A haronic osciator has a forc constant of 475 N an a ass of.6-7 kg. Cacuat th ZPE an th photon frqunc n to bring th osciator fro groun stat to th first cit stat. 7

8 k 475N s 7.6 kg N kg s E J 34 Js s E E 5.7 J E 5.7 J s h Js 3476n c Eap: Th IR spctru of H 35 C is oinat b a pak at 886 c -. Cacuat th forc constant of th ocu. Ruc ass au c~ k kg s 7 475N c s 886c kg 5.43 kg au 4 s s kg 8

9 What about DC? HC.89au.66 kg au 3.4 kg k 475kg s s kg HC 4 ~ 3.89 s 63c ~ c c s Isotop ffct is iportant to assign spctra ins. Rigi rotors Anguar ontu vctor r p I r fi whr th anguar vocit an ont of inrtia ar HC t I r Th rotationa nrg cassica: E I 9

10 Schröingr qn. for D otion: E Us poar coorinat sst sin cos r r it bcos for D rigi rotor: E I Gnra soution: IE i Ccic bounar conition: So i i i i i N N N has to b vn intgrs i.....

11 Quantization I E D Anguar ontu: z quantiz anguar ontu Anguar ontu oprator i i p p z is a coon ignfunction of both oprators: i i i i i z I I H i Uncrtaint princip just ik p an z

12 3D rotors: Sphrica coorinats: rsin cos rsin sin z rcos z r Haitonian: sin r H r r r r z Th iffrntia oprator is ca apacian. If r is fi th 3D rigi rotor Haitonian: H rigi rotor I whr

13 sin sin sin Schröingr qn. I E Th soution is th sphrica haronics: whr Y N P cos N!! i D rotor an P cos is th associat gnr function. Th quantization of two anguar coorinats as to two quantu nubrs: =... = ± ±... ± + 3

14 Rotationa nrg 3D rotor E I Sinc E is inpnnt of ach v is + gnrat which can b rov b an trna fi Zan ffct. Whn P cos bcos th gnr ponoia: P cos P cos cos 3cos P cos For P cos sin cos P cos Eaps of sphrica haronics: Y Y Y cos sin i Y 3cos 4

15 5 Th ar ignfunctions for both an z : Y Y z Y Y Vrification for cos Y cos sin sin sin sin cos sin sin sin I I H cos i z Th sphrica haronics ar orthonora Y Y * sin Not th vou nt an rangs.

16 Sphrica haronics hav nos u to gnr functions. For Y cos th anguar no can b trin b cos o 9 Vctor o: Th anguar ontu can b consir as a vctor. In quantu chanics it is quantiz in i. agnitu: ii. orintation: z On crtain irctions of ar aow. Eap = = ±: Eap: Cac. apitu of z an E for a rotor with I = kg an =. 6

17 Js Js z.55 Js 3-fo gn. 34 Th ang btwn an th z-ais for : cos z o 45 E I Js kg.4 J Eap. A 3D rotor absorbs at c -. Cacuat its ont of inrtia. E E E I E h hc ~ so I ~ hc c ~ 34.5 Js c s c kg 7

Chemistry 342 Spring, The Hydrogen Atom.

Chemistry 342 Spring, The Hydrogen Atom. Th Hyrogn Ato. Th quation. Th first quation w want to sov is φ This quation is of faiiar for; rca that for th fr partic, w ha ψ x for which th soution is Sinc k ψ ψ(x) a cos kx a / k sin kx ± ix cos x