Quantization of Atomic Energy Levels

Transcription

1 Quatizatio o Atomic Egy Lvls Atomic Scta Th ist al clus to th tu atu ad stuctu o atoms 1 w ovidd by atomic scta Dcads bo uthod dvlod his modl o th atom ad Plack advacd his quatum thoy o blackbody adiatio, it was kow that atomic gass (lmts) xhibitd disct o li scta Such gass, wh hatd to a high ough tmatu o a submittd to a lctical dischag, will mit visibl light Wh this light is assd though a diactio gatig o ism, o ids that oly ctai shaly did wavlgths a mittd I additio to th visibl wavlgths, th a iad ad ultaviolt lis as wll Ths wavlgths mak u th missio sctum o th atom o which th gas is comosd I whit light (cotiuous sctum o wavlgths) is assd though a cool atomic gas, th cotiuous sctum o th whit light is oud to b cossd by dak lis at sciic wavlgths This idicatd that ths sciic wavlgths w big absobd by th atoms o th gas Fo a sigl gas o atoms, th lis i th absotio sctum coicid xactly with lis i th missio sctum Hc, th atom mits light at th sam sciic wavlgths as it absobs light Moov, ach atomic gas was oud to hav a uiqu missio sctum, which suggsts that th sctum is coctd with th stuctu o th atom 3 As you call, uthod cocivd o th latay modl o th atom basd uo his alhaaticl ximts Howv, this modl atom is ihtly ustabl bcaus o th cotiuous missio o lctomagtic adiatio by th acclatig lctos I 1913, Nils Boh oosd a modl o th atom that was sctaculaly succssul i xlaiig th scta o o-lcto, o hydog-lik, atoms Th modl was basd uo uthod s modl, but i a bold st, Boh datd om th classical thoy o adiatio that was th dmis o th uthod modl Th lmts o Boh s modl ca b summaizd i ou ostulats (1) Th Coulomb (lctostatic) attactio btw a lcto ad th uclus ovids th ctital oc cssay o a dyamically stabl cicula obit [Classical] () Th mittd obits a thos i which th agula momtum o th lcto is a itg multil o ħ, i, L = ħ, wh = h / π ad = 1,, 3, [No-classical] (3) A lcto movig i a mittd stabl obit dos ot adiat [No-classical] (4) Emissio o absotio o adiatio occus oly wh a lcto maks a tasitio btw stabl obits [No-Classical] Postulat cam about as a sult o Boh s alicatio o th Cosodc Picil, which says that i th limit o lag obits ad lag gis wh classical hysics is valid, quatum 1 Mchaical Uivs Vido Atoms, Ch 0 Li h sts a imag o th slit usd i sctosco o sctomt 3 Show missio ad absotio scta 1

2 calculatios must ag with classical os I th macoscoic wold wh classical hysics holds, th agula momtum is a cotiuous vaiabl Statig om this gim, Boh was abl to show that th agula momtum was quatizd accodig to ostulat Th 3 d ostulat is a siml asstio that cicumvts th classical adiatio oblm Th 4 th ostulat cocts with Plack s quatum thoy i which gy lvls o atoms (oscillatos) a quatizd ad missio o absotio o adiat gy occus i disct amouts cosodig to th gy dic btw two lvls W ow utiliz th Boh modl to div a xssio o th gy o th lcto i a olcto atom Boh Thoy W assum th uclus has ositiv chag +Z ad is iiitly massiv (as a as th lcto is cocd) ad tho udgos o motio W shall discuss coctios o th iit mass o th uclus lat W also assum th lcto is olativistic - v +Z Equatig th lctic oc to th ctital oc givs kz mv =, (61) o, kz = mv (6) I th cdig, k = 1 4 πε0 1 Th kitic gy o th lcto K = mv which om Eq (6) is giv by K, 1 kz = mv = (63) Now, th lctic ottial gy U o th lcto is giv by U = kz Th total mchaical gy E o th lcto is th kz E = K + U = (64) Not that E is gativ bcaus th lcto is boud, i, gy must b sulid to mov th lcto om th atom Now to dtmi E, w d to id To do this, w ivok th scod ostulat: L = = 1,,3 (65) Fo a lcto i a cicula obit about th uclus, L = mv Thus, usig Eq (65), w id that

3 v = (66) m Substitutig o v i Eq (6) yilds o, kz = m m, 1,,3, = = mkz (67) Thus, th quatizatio o th agula momtum lads to th quatizatio o th obits Now, w ca wit ab = =, mkz Z wh a B = (68) mk is did to b th Boh adius Its valu is a B = 0059 m Substitutig o i th xssio o E yilds kz k 1 E = = Z, ab ab o, ( Z ) 4 1 mk E = Z, 1,,3, = (69) Hc, w s that gy quatizatio sults om th quatizatio o agula momtum Eq (69) alis to all o-lcto atoms, o xaml, hydog (Z = 1); H + (Z = ); Li + (Z = 3), ad so o Lt us sciically cosid th cas o hydog (Z = 1) Th, 4 1 mk E = 1,,3, ( Z 1) = = (610) Lt us di th ydbg gy: 4 mk E = (611) Thus, o hydog, th allowd gis a giv by E E 1,,3 = = (61) Th valu o th ydbg gy E = 136 V 3

4 Accodig to th 4 th ostulat, gy is mittd o absobd by th atom wh th is a tasitio btw gy stats I oth wods, a tasitio btw gy stats is accomaid by th missio o absotio o a hoto o gy cisly qual to th dic btw th gis o th two stats Cosid th missio ocss o hydog Th atom is iitially i a high gy stat ad maks a tasitio to a low gy stat, with a hoto big mittd By gy cosvatio, w hav hc Eh = h = = Ei E (613) λ Now, E E 1 1 Ei E = = E i i Fom Eqs (613) ad (614), w id 1 E 1 1 = λ hc i Lt us di th ydbg costat = E hc Th Eq (615) bcoms = ( Z = 1) λ i Th valu o th ydbg costat is m -1 (614) (615) (616) Eq (616) givs us th wavlgths o all th ossibl hotos that ca b mittd wh a hydog atom maks a tasitio om a high gy stat (lag ) to b a low gy stat (small ) I oth wods, it dicts th wavlgths o all th lis o th missio (ad absotio) sctum o hydog Th wavlgths dictd by Eq (616) a i xcllt agmt with ximtal sults Th lowst-gy stat o a atom is calld th goud stat Claly, i th Boh atom, th goud stat cosods to = 1 (most gativ gy) Fo hydog, th goud-stat gy is E1 = E 1 = 136 V Th = lvl is calld th ist xcitd stat Its gy is E = E = 34 V Th = 3 lvl is th scod xcitd stat, ad so oth Th atomic gy stats a ot schmatically dictd as a gy lvl diagam Each lvl cosods to th gy o a lcto stat Aows show th tasitios btw lvls Absotio o a hoto is idicatd by a = 4 aow oitig om a low to a high gy = 3 lvl Emissio o a hoto is idicatd by a aow om a high to a low gy lvl = Ths gy lvl diagams allow o to covitly dict dit sctal sis, which a gous o tasitios, ach o which cotais tasitios that tmiat at th sam = 1 4

5 gy lvl Fo xaml, th Balm sis, which cotais th wll kow visibl lis o th hydog sctum, cosists o tasitios i which th iitial gy lvls hav i > ad th ial gy lvl i all cass is = Fo th Balm Sis (missio), = (Balm Sis, hydog) (617) λ i Wh i = 3 i Eq (617), o ids that λ = 656 m, which cosods to th d li ducd Mass I th divatio o th gy o th lcto i a o-lcto atom, w mad th assumtio that th uclus is iiitly massiv ad dos ot mov Though this is a asoabl aoximatio giv th act that m m 000, sctoscoy is a vy cis scic ad th discacy i th wavlgth du to th iiit mass assumtio ca b dtctd Hc, to mak ou omulas mo accuat, w must accout o th iit mass o th uclus ad its motio W us oblm 51, Taylo, Zaiatos, ad Dubso, d d as a vhicl to accomlish this [Show diagam i class; diagam i th txtbook] Fist, w ot that th ct o mass (CM) o th hydog atom is uacclatd sic th t xtal oc o th atom is zo Hc, w ca lac th CM at st i a aoiat itial am Th oto ad lcto obit about th statioay CM W assum th oto ad lcto obit th CM igidly, so thy hav a commo agula vlocity ω Futh, lt us tak th oigi to b at th oto's ositio (a) Th ositio o th CM is giv by CM m1 1 + m = m + m 1 m (0) + m m = CM = = m + m m + m (i) m m = = 1 = m m m m (ii) (b) wh v = ω v = ω K = K + K = m ω + m ω m m mm 1 m m = ω + = ω ( m ) ( ) m m m m + m = µ ω, (iii) 5

6 is th ducd mass mm µ = m + m [Notic that th oly dic btw th abov xssio o K ad th ixd-oto sult is that µ has lacd m Also, ot that µ = 09995m o hydog] (c) Th lctostatic oc xtd o th oto is F = mv mω Th ctital oc = = = mω But So m = m + m m m ω m ω k / = = = µ ω m + m Equatig th lctostatic oc to th ctital oc givs k µ ω = (iv) Th sam sult is obtaid o th lcto (d) Th ottial gy is giv by k U = Usig Eq (iv), w ca wit k 1 1 U = = µ ω But K = µ ω Hc, K = U 1 k E = K + U = U = () L = L + L = m v + m v = m ω + m ω m m = + = ω m m µ ω m + m m + m () It is cla that coctig th viously divd sult oly tails lacig m by µ Thus, th ydbg gy bcoms 6

7 E µ k = 4 (618) Oth Evidc o th Quatizatio o Egy Lvls (a) Chaactistic X-ays W hav s that wh x-ays a oducd by bombadig a tagt matial with high-gy (kv) lctos, a cotiuous x-ay sctum is oducd (bmsstahlug) alog with disct aks that a chaactistic o th tagt matial Ths disct x-ay wavlgths a du to tasitios ivolvig th imost ( = 1,) lctos i th multilcto atom (S diagam blow) A bombadig lcto kocks out a i lcto, catig a vacacy Th atom is ow i a xcitd stat To tu to its lowst gy stat, a lcto om a high gy stat maks a tasitio to ill th vacacy at th low lvl This is accomaid by th missio o a x-ay hoto 4 Elctos at th = 1 lvl a taditioally calld K lctos Thus, a tasitio om th = lvl to th = 1 lvl is calld K α adiatio, om th = 3 lvl to th = 1 lvl is calld K β adiatio, tc Th gou o tasitios tmiatig at th K ( = 1) lvl is calld th K sis Tasitios that tmiat at th = lvl om th L sis, ad so oth Fom th Boh modl, o would dict that hoto gis ad qucis a ootioal to Z 1 1 [Fo a o-lcto atom with ucla chag +Z, E h Z E ( ) h i = = ] Mosly oud that th K α ad K β adiatio 5 o may lmts could b dscibd by th latio = a( Z δ ), (619) 4 Actually, a succssio o tasitios occu as vacacis a catd at high ad high gy lvls W a oly cocd with th lowst lvls, howv 5 Diagam o Mosly s data, 9 ohl 7

8 wh a is a costat ad δ 1 [Not that th x-ay lis a ot sigl lis, but closly sacd lis Th high lvls a slit by th si-obit itactio] Th ( Z 1) ddc occus bcaus th chag o th uclus is atially scd 6 Such scig cts a ot accoutd o i th Boh thoy Howv, Mosly s wok std stog vidc o th quatizd atu o atomic gy lvls, as dictd by Boh Fack-Htz Eximt This ximt std additioal coimatio o th quatizd atomic gy stats dictd by th Boh thoy Mcuy vao is bombadd with lctos o kow gy I th kitic gy (KE) o th lctos i lss tha th gy quid to xcit a Hg atom om th goud stat to th ist xcitd stat (ist xcitatio gy), th lctos mak oly lastic collisios with th massiv Hg atoms ad vy littl KE is lost by th lctos (1 at i 10 5 ) Howv, wh th KE o th lctos is qual to th ist xcitatio gy, a lcto ca giv u all o its KE to a Hg atom i a ilastic collisio, which xcits th Hg atom to th ist xcitd stat By comaig th KE o th lctos bo ad at collisio, o ca id th ist xcitatio gy [Show diagam o aaatus ad gah o lat cut vs gid voltag om txtbook] Th F-H aaatus cosists o a Hg tub cotaiig a cathod, a lat (collcto), ad a acclatig gid Both th lat ad gid a hld at a ositiv ottial with sct to th cathod, but th lat ottial is slightly low ( 05 V) tha th gid This tadig ottial vts lctos with gligibl KE om achig th lat ad thby cotibutig to th lat cut Thus, wh lctos mak ilastic collisios with Hg atoms i th viciity o th gid, th tadig ottial vts thos lctos with gligibl KE at collisio om achig th lat, ad th lat cut dos Fo Hg, such a do i lat cut occus at a acclatig voltag o 49 V, ad itg multils o this valu Th dos i cut at 49 V, 3 49 V, tc, a du to o lcto suig two o mo ilastic collisios with Hg atoms bo achig th gid, wh it udgos a ial ilastic collisio ad tho caot ach th lat Hc, th ist xcitatio gy o Hg should b 49 V This cosodd cisly with a 536-m sctal li that aad i th missio sctum o th vao Sic th ilastic collisios oly occu wh th bombadig lcto has KE gat ough to suly th 1 st xcitatio gy to a Hg atom, o cocluds that th gy stats a idd disct, as dictd by th Boh thoy Limitatios o th Boh Modl Though th Boh modl was vy succssul i xlaiig th scta o hydog-lik atoms, it aild to xlai th scta ad oth otis o atoms with mo tha o lcto Th 6 Each lcto i th K shll atially shilds o scs th ucla chag o th oth lcto i th K shll 8

9 aso is that th Boh modl could ot oly accout o th itactios amog th lctos W shall s that, i od to dal with multilcto atoms, th wav atu o th lctos has to b ivokd This ally dos ot dtact om th succss o Boh modl, sic its ultimat vlatio that atomic gy lvls a quatizd is still valid 7 7 Oth limitatios o Boh modl: (1) itsity o sctal lis; () Why is L quatizd? 9