Exam 2 is Wed. March 28, at 9:05-9:55 AM. Rooms: Last names beginning A - M Akerman 225; N Z Tate 110.

Transcription

1 hm 5 Rviw Sht fr Em Pstd //8 Rmidr: Em is Wd. Mrch 8, t 9:5-9:55 AM. Rms: Lst ms giig A - M Akrm 5; N Z Tt. It will cvr th mtril cvrd i lctr thrgh ri. Nv., d th rdigs d prlms : Pstlts d pricipls f qtm mchics, ctid prts t cvrd Em hptr, pgs 5-, Sctis. thrgh.6; Hmwrk 5 ttm f pg Rigid rttr hp. 5 sctis 8-9, Hmwrk 6 d Prlm St Hydrg tm hp. 6 sctis -7, Hmwrk 7. This DOES icld hp. 6 scti 7 p. 9, rif itrdcti t th H tm. Nt: NOT icldd r Em is hptr 7 Apprimt Mthds, lthgh rdig d prlms this mtril r ls icldd i Hmwrk 7. Mthhptr glr mmtm vctrs, icldd i Hmwrk 6 Mthhptr D sphricl crdits, icldd i Hmwrks 6 d 7 S pgs - fr th qtis tht will prvidd Em. S pgs -7 fr spcific tpics tht my icldd Em. Y my rig -prgrmml, -grphig clcltr. N ts r llwd. Pls rig yr U crd t th m; th prctr my spt-chck ths wh cllctig th ms. A prctic Em d its swr ky will pstd fr prctic t wk fr r m.

2 Pssily Usfl Eqtis, vrsis d stts hm 5 c m/s k B.695 cm - / K h 6.66 J s! h/.55 J s.6 9 V.6 9 J crrspds t 866 cm - m 9.9 kg m p.67 7 kg m.66-7 kg H tm i cs i si ν ~ 9,678 / - / cm - E - m r ε h 8ε h m lssicl wv qti,t / /υ,t / t Nrml mds f virtig strig f lgth l:,t A cs ω t φ si / l Schrödigr qti:! / m d Ψ / d VΨ EΨ Mmtm prtr: P -i! / PIB ψ / ½ si / E h / 8m HO - / ψ whr µk ½ / ¼ /! E v v½ hν whr ν / k/µ ½ d µ m m / m m ditmic Rttil mti: K ½ Iω L / I I µr L Iω mrv Rigid rttr: E J! /I JJ E cm - B JJ B Hz h / 8 I B cm - h / 8 ci ω v/r Hydrg tm: ν ~ 9,678 / - / cm - E - m V 8ε h ε ε h.59 Å m ψ s / ½ / / - r / Φ m φ / ½ im φ Aglr mmtm: L ħ l l L z m! L z - i ħ ˆ / φ Sphricl crdits: dv r siθ dr dθ dφ

3 i i B A slti is k d d B A slti is k d d Eqtis Diffrtil d d d gr psitiv d d d d d d d d d d d d d d d d d : si cs : it! cs cs si si si si si si si si si 8 cs si si si cs si si cs si cs si si cs si l

4 Pstlts f Qtm Mchics hptr, pp. 5- Effct f msrmt chgig th stt f th prticl Orthrml wv fctis; Krckr dlt mmttr f tw prtrs d th crtity pricipl Pstlt 5: rcgiz th tim-dpdt Schrödigr qti, d its slti p. 6 q... Usig this prssi, l t shw tht if systm is i sttiry stt tht is, igfcti f th Hmilti prtr, th its prility dsitis d vrgs f srvl prprtis will t vry with tim s p. 6 qti.. Rttil Mti "Rigid Rttr" Mdl hptr 5, Sctis 8-9 Rviw f clssicl rttil mti; crrspdcs tw lir systms d rttig systms mmt f irti, ctr f mss, rttil spd, glr mmtm, kitic rgy csrvti f glr mmtm t rttil rgy t csrvd fr ic sktr vctr rprstti f glr mmtm; L r p; right-hd rl s pp. - Rigid rttr: mdl fr gs phs mlcl rttig frly i spc t its ctr f mss mst csidr -D rtti rthr th plr mti i viw f crtity pricipl writig th Schrödigr qti sphricl crdits: dfiitis d rgs f th thr vrils Lplci prtr i rtsi crdits t: this is giv i sphricl crdits th qti list qtm mrs: J,,,... dtrmis th llwd rgis d th mgitd f th glr mmtm m, ±, ±,... ±J dtrmis th prjcti f th glr mmtm th z is diffrt m stts fr giv J r dgrt hv sm rgy i sc f trl fild s th dgrcy g f giv J stt is J wv fctis fr th rigid rttr r th sphricl hrmics, Yθ,φ lwst rgy stt is sphriclly symmtric: Y / ½ E J, m, rtti rigid rttr rgy lvls: E J cm - Bcm - JJ cm - is rlly rgy dividd y hc Micrwv spctrscpy: slcti rls: ΔJ ±; prmt dipl mmt if mlcl ds't hv prmt dipl mmt, it still hs rttil lvls, t lctrmgtic rditi ds t idc chg i th rttil stt micrwv spctrm shws sris f lis spcd y B cm - sig micrwv spctrscpy r th spcig tw rttil rgy lvls t clclt th d lgth f ditmic mlcl, d vic vrs drwig th rttil lvls d th virtil lvls f ditmic mlcl sm grph, pprimtly t scl t: thr r rttil lvls lw th v virtil lvl, d th rttil lvls r mch mr clsly spcd th th virtil lvls - s p. 99 ig..

5 similritis d diffrcs tw th clssicl d th qtm mchicl rigid rttrs; pplicti f th "crrspdc pricipl" t vry high qtm mrs, th qtm prdictis pprch th clssicl prdictis t rttil rgis: As J, th rgy diffrcs tw sqtil J lvls dividd y th ttl rttil rgy pprch zr tht is, rgis r sstilly ctis. Aglr mmtm: sqr f glr mmtm dpds J sm s fr l fr th H tm Hydrg Atm hptr 6, Sctis - 7 Writig th Schrödigr qti fr -lctr tm r i might icld clr chrg, Z writ K, th kitic rgy prtr; V, th pttil rgy prtr; H, th Hmilti prtr hw t clclt µ, th rdcd mss, i this prlm why w s dimsis rthr th ssmig plr mti Us f sphricl crdits LPlci prtr, vlm lmt t: th f ths r giv th qti sht why w s sphricl rthr th rtsi crdits fr this prlm Eprssig th ttl wv fcti "ritl" s prdct f fctis which ch dpd sigl vril: ψr,θ,φ Rr Θθ Φφ c d this cs th Hmilti prtr is sprl i sphricl crdits Rltig th vl f sprti cstt β t th rgy f th rigid rttr Aglr prts f th wv fctis Yθ,φ, "sphricl hrmics" d glr mmtm prprtis: Yθ,φ r th sm s th wv fctis fr th rigid rttr s th tw systms H tm d RR shr th sm glr mmtm prprtis w s th syml "l" fr th H tm i plc f "J" fr th rigid rtr Φφ: strtig with th φ-dpdt prt f th Schrödigr qti, slv fr Φφ icldig pplyig dry cditis d tiig th rmlizti cstt Θθ iclds fcti f cs θ sscitd Lgdr fcti; dpds th l d m fr l p ritl d m, this is jst cs θ Qtizti f th mgitd f th glr mmtm: L! ll; L! ll ½ th sphricl hrmics d th vrll H tm wv fctis r igfctis f th prtr L, with igfctis! ll llig th ritls s s, p, d, f ccrdig t th vl f l,,,, rspctivly Qtizti f th dircti f th glr mmtm vctr spc qtizti! : m dtrmis th prjcti f th glr mm. spc-fid is "z": L z m! th sphricl hrmics r igfctis f prtr L z -i! / φ, with igvls mħ Vctr rprstti f th lgth f th glr mmtm vctr d its dircti cs Aglr mmtm vctr ct ligd lg th z is crtity pricipl L z is lwys lss th L 5

6 Ucrtity rltis mg prtrs fr L, L, L y, L z which f ths prprtis c wll-dfid simltsly Nt: th qti fr L z is th qti sht; if y d thr glr mmtm prtrs, thy wld prvidd. Rcll p. : If tw prtrs cmmt, th crrspdig prprtis c th msrd simltsly with crtity If tw prtrs d t cmmt tht is, th rdr f prtis mttrs, th crrspdig prprtis sch s th d z cmpts f th glr mmtm r "cmplmtry srvls" r rltd y crtity rlti. strcti f th p d p y ritls s lir cmitis f th m ± ritls cstrcti f th d ritls s lir cmitis f ritls with m± r ± p d p y ritls d t hv wll-dfid vls f m thy r t igfctis f L z Rdil prts f th wv fctis, Rr: dpd qtm mrs d l ivlv prdct f trms, icldig: -r /, s ritls with lrgr r mr pdd rl, s if l>, th wv fcti gs t zr t th cls plymil i r sscitd Lgrr fcti dpds th d l, -l- # rdil ds vls f r t which Rr, thr th r d r Slvig S. qti fr l t fid th lwst rgy grd stt wv fcti d rgy; hvir f Rr t lrg r dry cditis Schrödigr qti with glr mmtm; csqcs f "ctrifgl trm"! ll/µr w Rr gs t zr t th cls B l t idtify th ritl s, s, s, p, p, d, tc. frm plt f Rr, r f Rr, r f th rdil distriti fcti r R s p. s fcti f r, r frm th hvir f th fcti t r d th mr f ds. Or, l t plt ths qlittivly fr giv ritl. s frm th rdil distriti fcti: fr, fr mpl lctr i s ritl hs highr prility f ig cls t th cls th lctr i p ritl. This css shildig ffcts i mltilctr tms, s s cms lwr i rgy th p. Tlig: clssiclly llwd d fridd rgis fr giv qtm stt i th s ritl, th lctr hs sigifict % prility f ig tsid th clssiclly llwd rgi > frm th cls, th "clssicl trig pit" hw t clclt th clssicl trig pit hw t cct fr this iqly qtm mchicl phm which thr systms tht w hv stdid prvisly shw tlig hvir Hw t clclt th fllwig r-dpdt prprtis fr giv wv fcti: vrg vl f r r f diffrt prprty tht dpds r prility pr it vlm f fidig th lctr distc r frm th cls prility f fidig lctr tw r d rdr frm th cls rdil distriti fcti Pr r Ψ mst prl distc, r, t which t fid lctr frm cls th vl f r withi which thr's giv prility f fidig lctr.g., 9% 6

7 Ovrll wv fctis, which r prdcts f th rdil d glr prts: Mig f qtm mrs, l, m; wht prprtis d thy dtrmi wht r th pssil vls f ch f ths qtm mrs mr f glr d/r rdil ds i giv wv fcti Dgrcy mr f ritls with sm rgy : fr giv shll r rgy -R H /, dgrcy fr giv d l sshll, dgrcy l B l t: idtify ritl frm its prility dsity plt, s p. drw th glr prt f th wv fcti fr s, p r d ritl rl rprstti idtify th mr f rdil r glr ds, giv wv fcti r st f qtm mrs Sigificc f th sig phs f wv fcti Ergy lvls: digrm, spctrscpic trsitis missi, srpti, iizti rgy ths rslts r th sm s i hptr - Rydrg frml Bhr mdl: Similritis d diffrcs tw th Bhr mdl f th H tm d th prst mdl Hlim tm: Writig th Schrödigr qti fr th hlim tm pg 9 Why c this qti NOT slvd ctly? 7