Quantum Mechanics & Spectroscopy Prof. Jason Goodpaster. Problem Set #2 ANSWER KEY (5 questions, 10 points)

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1 Chm 5 Problm St # ANSWER KEY 5 qustios, poits Qutum Mchics & Spctroscopy Prof. Jso Goodpstr Du ridy, b. 6 S th lst pgs for possibly usful costts, qutios d itgrls. Ths will lso b icludd o our futur ms.. poits.5 poits Stt th Corrspodc Pricipl. Th qutum mchicl rsults td to gr with th clssicl prdictios i th limit of vry high qutum umbrs. S p. 86 b.5 poits Giv two mpls of th pplictio of th Corrspodc Pricipl to th prticli--bo systm.. At lrg, th spcigs btw ods d btw mim i th wv fuctios dcrs, so th prticl is sstilly qully likly to b foud ywhr i th bo, s would b th cs clssiclly.. At lrg, th spcig btw coscutiv rgy lvls bcoms smll s comprd with th totl rgy, pprochig th clssicl prdictio of cotiuum of rgy lvls. or mpl: E - E - / E - - / - - / - / / / This rtio pprochs s pprochs.

2 . poits.75 poits St up th itgrl tht you would us to clcult th vrg vlu of th positio of prticl of mss m i o-dimsiol bo of lgth with wlls t d, if th prticl is i Ψ i.., th stt. b.75 poits Solv th itgrl to obti th vrg vlu. Show ll of th stps i your clcultio. Not: thr my b usful itgrl o th lst pg. c.5 poit Could you hv prdictd th vrg vlu simply by cosidrig th symmtry of th wvfuctio squrd with rspct to /? Epli. Ys, bcus th squr of th wvfuctio ψ is symmtricl bout th middl of th bo /. Tht is, it is idticl for through / d for / through rgrdlss of th vlu of. So, th vrg vlu of th prticl's positio,<>, is t th middl of th bo.

3 . poits poits St up th prssio tht you would us to clcult th probbility tht prticl of mss m i th groud stt of o-dimsiol bo of lgth with wlls t d will b foud- btw d.. Substitut i ll vlus, but do ot solv. b poits Solv th itgrl d clcult th probbility. Show ll of th stps i your clcultio. Not: thr my b usful itgrl o th lst pg.

4 . poits Th fr-lctro modl for th lctroic stts of lir cojugtd hydrocrbos polys uss th o-dimsiol prticl-i--bo pproimtio for th rltiv rgis of th d * molculr orbitls. or htri C 6 H 8, which hs si lctros, th ovrll lgth of th molcul c b ssumd to b bout 9. Å 6. - m. Prdict th photo rgy rquird to cit lctro from th HOMO highst occupid molculr orbitl to th * LUMO lowst uoccupid molculr orbitl E photo ΔE systm J J

5 5. poits or ch of th systms tht w will discuss this smstr, it is usful to compr th swrs to th followig qustios. Hr, swr ths qustios for th o-dimsiol prticl-i-bo systm. This is from rviw Homwork.. Wht is this systm modl for tht is, wh do w us it? Giv two mpls.. Writ th Schrödigr qutio dfi ll of your trms, icludig digrm dfiig th vribls or coordits usd.. Writ th pottil rgy oprtor for this systm.. Writ th kitic rgy oprtor for this systm. 5. Wht qutum umbrs r ivolvd d wht r thir possibl vlus? 6. Wht proprtis r "wll-dfid", if th systm is i igstt of Ĥ, th Hmiltoi oprtor tht is, if th systm is i "sttiory stt"? By "wll dfid", w m tht th rsults of giv idl msurmt c b prdictd with crtity. 7. Drw pictur rprstig th wv fuctios of th groud stt d of th first citd stt. 8. Dos th groud stt disply tulig bhvior? or th prticl-i--bo, this ms - c th prticl b foud outsid th bo? 9. Writ th formul for th rgy lvls ths r th igvlus of Ĥ.. Writ th formul for th sttiory stt wv fuctios ths r th igfuctios of Ĥ.. Writ th formul for th groud stt wv fuctio.. Writ th formul for th rgy ssocitd with th groud stt wv fuctio tht is, th miimum rgy.. Writ th formul for th cm - /λ of photo bsorbd i th trsitio from th groud stt to th first citd stt.. Dscrib th dgrcy - i.., th umbr of stts with th sm rgy for th o-dimsiol prticl-i--bo. Not: if thr is oly o stt t ch rgy lvl, th th dgrcy is just. s t pg for swrs 5

6 . Dgrcy g tht is, thr is oly o stt t ch rgy lvl, sic thr is oly o qutum umbr for th -dimsiol prticl-i--bo 6

7 Possibly Usful Equtios, Covrsios d Costts Chm 5 c m/s k B.695 cm - / K h 6.66 J s! h/.55 J s.6 9 C V.6 9 J corrspods to 866 cm - m 9.9 kg m p.67 7 kg mu.66-7 kg H tom i cos i si ~ ν 9,678 / - / cm - E - m r ε h 8ε h m Clssicl wv qutio u,t / /υ u,t / t Norml mods of vibrtig strig of lgth l: u,t A cos ω t φ si / l Schrödigr qutio:! / m d Ψ / d VΨ EΨ Momtum oprtor: P -i! / PIB ψ / ½ si / E h / 8m cotiud o th t pg 7

8 8 k k ik ik b b b b o o o B A solutio is k d d C C k B k A solutio is k d d Equtios Diffrtil b b d b b d d gr positiv d d d d d d d d d d d d d d d C u u du C u du u : si cos : it! cos cos si si si si si si si si si 8 cos si si si cos si si cos si cos si si cos si l

Chem 4502 Prof. Doreen Leopold 10/18/2017 Name (Please print) Quantum Chemistry and Spectroscopy Exam 2 (100 points, 50 minutes, 13 questions)

Chem 4502 Prof. Doreen Leopold 10/18/2017 Name (Please print) Quantum Chemistry and Spectroscopy Exam 2 (100 points, 50 minutes, 13 questions) Chm 5 Prof. Dor Lopold ANSWER KEY /8/7 Nm Pls prit Qutum Chmistry d Spctroscopy Em poits, 5 miuts, qustios Pls chck hr if you would prfr your grdd m to b rturd to you dirctly rthr th big icludd mog lphbtizd