1. Can not explain why certain spectral lines are more intense. 2. many spectral lines actually consist of several separate lines

Transcription

1 Chatr 5 Quatu Mchacs ltato of Bohr thory:. Ca ot la why crta sctral ls ar or ts tha othrs.. ay sctral ls actually cosst of svral sarat ls whos λ dffr slghtly. 3. a udrstadg of how dvdual atos tract wth o aothr to fro acroscoc attrs Quatu chacs(95~96) 5. Quatu Mchacs Classcal chacs futur hstory of a artcl s coltly dtrd by ts tal osto & otu togthr wth forc. Quatu Mchacs suggst th atur of a obsrvabl quatty ucrtaty rcl robablts classcal chacs s a aroat vrso of quatu chacs Wav fucto φ. robablty of fdg th body for col φ =

2 φ*φ * (φ * : col cojugat) φ=a+ιb φ * =A-ιB φ * φ=a +B wll bhavd wav fucto () φust b cotuous & sgl-valud vrywhr (),, ust b sgl valud & cotuous(for y z otu cosdrato) (3) φust b oralzato, whch as thatφ ust go to 0 as y z dv ds to b a ft costat = robablty dsty P dv dv oralzato robablty d a artcl a bo, φ=0 outsd th bo but ral cas, vr ha.

3 5. wav quato y v y t soluto: y=f(t /v) cosdr a wav quvalt of fr artcls. Y=A - ω (t- /v) {udad(costat altud A), oochroatc( cost ω), haroc} Y=Acosω(t /v)-asω(t /v) For a strchd strg, oly ral art has sgfcac. 3

4 4 5.3 Schrodgr s quato : t ddt for for a fr artcl ) ( ) / ( t v t A A )...(, ) ( A A h h t h urstrctd artcl of rgy & otu P ovg + drcto () dffrtatg q(a) for φ twc wth rsct to () dffrtatg q(a) forφwth rsct to t t t for v<<c U rstrcto U t U t U ), ( Drvd fro fr artcl, but t s a gral cas. If U ow φca b solvd.

5 5.4 ctato valu calculatd th ctato valu <> Th valu of w would obta f w asur th ostos of a grat ay artcls dscrbd by th sa wav fucto at t t ad th avrag th rsults. Th avrag osto of a ubr of dtcal artcls dstrbutd alog as. N at, N at ; N N N33... N N N... 3 N N If alog wth a sgl artcl, rlacd N by robablty P d d d If φ s a oralzd fucto d ctato valu of osto G d G d 5

6 5.5 Schrodgr s quato: stady-stat for for o-dsoal wav fucto Ψ of a urstrctd artcl ay b wrtt t t t A A Ψ s th roduct of a t-ddt fucto t ad a osto-ddt fuctoψ If Ψ=F() F'(t) Th t varatos of all wav fuctos of artcls actd o by statoary forcs hav th sa for as that of a urstrctd artcl. t substtutg to t-ddt q U t t t t U U 0 Stady-stat Schrodgr q -D U 0 y z 3-D 6

7 ** For Schrodgr s stady-stat q, f t has o or or solutos for a gv syst, ach of ths wav fuctos corrsods to a scfc valu of rgy. rgy quatzato Cosdrg stadg wavs a strtchd strg of lgth that s fd at both ds. ths wavs ar subjct to th codto(boudary codto) that y=0 at both ds. φ & d to b cotuous, ft, ad sgl-valu λ =/+, =0,,,3 cobato of wav q & boudary codto. y(,t) ca st oly for crtaλ gvalus & gfuctos Th valu of rgy for whch Schrodgr s stady-stat q ca b solvd ar calld gvalus ad th corrsodg wav fuctos φ ar calld gfuctos. Th dscrt rgy lvls of H ato 4 3 o,,,3,... 7

8 ar a al of a st of gvalus. I addto to, agular otu s also quatzd. I H ato, th gvalus of th agtud of th total agular otu ar l, l 0,,,3...( ) A dyac varabl G ay ot b quatad. asurts of G ad o a ubr of dtcal systs wll ot yld a uqu rsult but a srad of valus whch avrag s ctato valu. G G d for al, H ato, osto s ot quatzd. 8

9 5.6 artcl a bo U(0<<)=0=costat φs 0 for 0 & d d th oto of a artcl s cofd btw =0 & = by ftly hard wall(t U(0)=U()= ) A artcl dos ot los rgy wh t collds wth hard walls. wth th bo: 0( U 0)...(5.4) q(5.4) has th soluto As Bcos B.C. φ=0 at =0 & = cos0= B=0 ( φ(=0)=0) ) 0 ( =,,3,. rgy of artcl ca hav oly crta valus gvalus rgy lvls = =,,3. As As ( gfuct os) 9

10 ths gfucto t all rqurts φ s a ft, sgl-valud, ad & cotuous (ct at th ds of th bo) To oralz φ 0 d d A 0 0 A A d 0 s d cos d A s s,,, A ( s cos ) 0

11 * s φ ay b -, but s + ( s robablty dsty of fdg th artcl) *wh =, th artcl ost lly to b th ddl of th bo but wh =, =0 th ddl of th bo. 5.3 Fd th robablty that a artcl trad a bo wd ca b foud btw 0.45 & 0.55 For = & =

12 Classcally, w ct th artcl to b ths rgo 0% of th t ( ddg o ) but QM gvs dffrt rdcto P, for =, P =9.8% =, P =0.65% s d d s

13 3 5.4 Fd <> of th osto of a artcl trad a bo wd <>= 4 8 cos 4 s 4 : 0 d Mddl of th bo!!

14 5.7 ft ottal wll *Pottal rgs ar vr cosdr ottal wlls wth barrrs of ft hght *Partcl rgy <U classcal chacs: wh artcl strs th sd of th wll, t boucs off wthout trg rgos I QM, t has a crta robablty of tratg to rgos Ⅰ&Ⅲ *I Ⅰ&Ⅲ U 0 d a 0 d U a a A B a <0, > φⅢ = C a +D -a -a wh - a wh B=C=0 φⅠ=a a, <0 φⅢd -a,> 4

15 ** ths wav fuctos dcras otally sd th barrr. Wth th wll φⅡ s F cos φ s cotuous φⅠ(=0)=φⅡ(=0) A=F φⅡ(=)=φⅢ(=) = D -a solv ( 0) at =0 & = s cotuous cobg ths B.C. solv colt wav fucto **Bcaus th wavlgths that ft to th wll ar logr tha for a ft wll of th sa wdth artcl otu ar lowr ( P=h/λ) ar lowr tha thy ar for a artcl a ft wll Th wav fucto trats th walls, whch lowrs th rgy lvls. 5

16 5.8 Tul ffct Partcl strs a ottal U(<U) th barrr has ft wdth (s Fg 5.8) artcl has o-zro robablty to ass through th barrr & rg o th othr sd. : tul dod: ' ass through ottal barrr v though thr K<barrr hght I rgo Ⅰ&Ⅲ U=0 d 0 d d 0 d A F B G cos s = (q 5.43) cos s q 5.43 th sa as artcl a bo A rrsts cog wav B rrst rflctd wav 6

17 φⅠ=φⅠ+ +φⅠ- φⅢ+ = F rrstd trasttd wav rgo Ⅲ othg could rflct th wav G 0 φⅢ=φⅢ+= F 7

18 v= s th grou vlocty of cog wav (qual to v of artcls) S v s th flu of artcls that arrvs at th barrr, S= # of artcls/ sc ( Trassso robablty # 3 sc ) T v v FF * v AA * v d d classcally T=0 <U I rgo Ⅱ Sch q C U 0 D, U (sa as ft ottal wll) ar ral quatts φⅡ dos ot oscllat ad artcl ay rg to Ⅲ or rtur toⅠ s ot zro 8

19 9 alyg B.C. & d to b cotuous at =o d d d d (s Fg 5.9) at = φⅡ=φⅢ dφⅡ/d =dφⅢ/d A+B=C+D D C F D C D C B A F A ) ( 4 4 t s assu U>> ) ( U also assu s wd ough >> F A F A 4 * 4 Hr vⅢ + =vⅠ - vⅢ + /vⅠ - =

20 0 h T aroato U U FF AA v AA v FF T / / 4 6 * * * *

21 5.9 Haroc oscllator Haroc oto: th rsc of a rstorg forc that acts to rtur th syst to ts qulbru cofgurato wh t s dsturbd. I th scal cas, th rstorg forc F follow Hoo s law d dt d dt h F=- -= 0 Acos t frqucy of haroc oscllator A: altud Φ: has agl dds o what s at t=0 I ost of cass, rstorg forcs do ot follow Hoo s low, but wh oly cosdr a sall dslact of rstorg forc ca b rcsd by Hoo s low. Ay syst whch sothg cuts sall vbratos about a q osto bhavs l a sl haroc oscllator.

22 .Maclaur s srs F()=F >0 + ( df ) =0 X + /( ) d =0 X 3 + /6( 3 d d d F d F ) =0 X 3 + =0 s q osto F=0=0 for sall, 3 s uch sallr tha F()=(dF/d)=0 X for rstorg forc (df/d)=0 s gatv Hoo s law.ottal rgy U()=- 0 F()d= 0 d=/.sch q y + /h (-/ )φ=0 (5.75) lt c=(/h h ) /, y=(/h h ) / =c = y y y c y y y y y = c c q5.75 c +/ (-/ )φ=0 +/ ( y y / φ)- / φ=0 lt α=/h( / ) y Sch q (5.75) y for ths q, wh y φ 0 for dy=

23 *for(5.78) oly wh α=+ =,=,=3 ca satsfy all codtos α=/h( / )=/hν & α=+ =(+/)hν =0,=,= rgy lvls of Haroc oscllator Zro ot rgy 0=/(hν) wh T 0 0 ot 0 3

24 H ato A artcl a bo A haroc oscllator qually sacd & o 0 4

25 for haroc oscllator ach α φ 4 y! H y φ cossts of a olyoal H(y) (Hrt olyoal) s tabl 5. Fro Fg 5., th artcl s abl to trat to classcally forbdd rgos wth a otally dcrasg robablty. 5

26 Classcal: a at d QM: Ma at ddl =0 QM: =0 Wh QM classcal Wh trato Orators, gfuctos & gvalus Is d?? d?? φ=φ(,t), I ordr to carry out th tgratos w d to rss P& as fuctos of,t but & o fucto as (,t)&(,t) Itgrato for s ot sutabl for <P> <> 6

27 7 for fr artcl t A K K U K t P t t orator d d U t U t sch q d d t d = d t ctato valu of a orator d G G, gvalu q G G Haltoa orator H U H

28 8 *Partcl a bo s d d cos d d d d 0 cos s 0 d otu gvalu ± as that th artcl s ovg bac & forth avrag 0 av * Fd otu gfucto d d s φ s ot otu gfucto d d s cos s

29 9 otu gfucto Varty & ar gfucto c d d ad ca b slar