General Physics (PHY 2140)

Transcription

1 Gnral Pysics (PHY 140) Lctur 16 Modrn Pysics Last lctur: 1. Quantum pysics Wav function Uncrtainty rlations Ligtning Rviw ΔΔ x p π ΔEΔt π Atomic Pysics Early modls of t atom Atomic spctra Bor s tory of ydrogn D Brogli wavlngt in t atom Quantum mcanics and Spin Rviw Problm: If mattr as a wav structur, wy is tis not obsrvabl in our daily xprincs? Captr 8 ttp:// 1 λ(basball) m 7.9 T Uncrtainty Principl Wn masurmnts ar mad, t xprimntr is always facd wit xprimntal uncrtaintis in t masurmnts Classical mcanics offrs no fundamntal barrir to ultimat rfinmnts in masurmnts Classical mcanics would allow for masurmnts wit arbitrarily small uncrtaintis T Uncrtainty Principl Quantum mcanics prdicts tat a barrir to masurmnts wit ultimatly small uncrtaintis dos xist In 197 Hisnbrg introducd t uncrtainty principl If a masurmnt of position of a particl is mad wit prcision Δx and a simultanous masurmnt of linar momntum is mad wit prcision Δp, tn t product of t two uncrtaintis can nvr b smallr tan /4π 3 4 1

2 T Uncrtainty Principl Matmatically, ΔxΔp x 4π It is pysically impossibl to masur simultanously t xact position and t xact linar momntum of a particl Anotr form of t principl dals wit nrgy and tim: ΔEΔt 4π Tougt Exprimnt t Uncrtainty Principl A tougt xprimnt for viwing an lctron wit a powrful microscop In ordr to s t lctron, at last on poton must bounc off f it During tis intraction, momntum is transfrrd from t poton to t lctron Trfor, t ligt tat allows you to accuratly locat t lctron ctron cangs t momntum of t lctron 5 6 Problm: macroscopic uncrtainty A 50.0-g g ball movs at 30.0 m/s. If its spd is masurd to an accuracy y of 0.10%, wat is t minimum uncrtainty in its position? A 50.0-g g ball movs at 30.0 m/s.. If its spd is masurd to an accuracy of 0.10%, wat is t minimum uncrtainty in its position? Givn: v = 30 m/s Δv/v = 0.10% m = 50.0 g Notic tat t ball is non-rlativistic. Tus, p = mv, and uncrtainty in masuring momntum is δv Δ p= m( Δ v) = m v v 3 = kg m s = kg m s ( )( ) Tus, uncrtainty rlation implis Find: δx =? J s Δx = = π ( Δ p) π ( kg m s) 3 m Too small to masur!! 7 8

3 Problm: Macroscopic masurmnt A 0.50-kg block rsts on t icy surfac of a frozn pond, wic w can assum to b frictionlss. If t location of t block is masurd to a prcision of 0.50 cm, wat spd must t block acquir bcaus of t masurmnt procss? Rcall: ΔxΔp and x p = mv 4π Atomic pysics: Captr Importanc of Hydrogn Atom Early Modls of t Atom Hydrogn is t simplst atom T quantum numbrs usd to caractriz t allowd stats of ydrogn can also b usd to dscrib (approximatly) t allowd stats of mor complx atoms Tis nabls us to undrstand t priodic tabl T ydrogn atom is an idal systm for prforming prcis comparisons of tory and xprimnt Also for improving our undrstanding of atomic structur Muc of wat w know about t ydrogn atom can b xtndd to otr singl-lctron lctron ions For xampl, H + and Li + J.J. Tomson s s modl of t atom A volum of positiv carg Elctrons mbddd trougout t volum A cang from Nwton s modl of t atom as a tiny, ard, indstructibl spr 11 watrmlon modl 1 3

4 Exprimntal tsts Early Modls of t Atom Expct: Rutrford s s modl 1. Mostly small angl scattring. No backward scattring vnts Rsults: 1. Mostly small scattring vnts. Svral backward scattrings!!! Plantary modl Basd on rsults of tin foil xprimnts Positiv carg is concntratd in t cntr of t atom, calld t nuclus Elctrons orbit t nuclus lik plants orbit t sun Problm: Rutrford s s modl T siz of t atom in Rutrford s modl is about m. (a) Dtrmin t attractiv lctrical forc btwn an lctron and a proton sparatd by tis distanc. (b) Dtrmin (in V) t lctrical potntial nrgy of t atom. T siz of t atom in Rutrford s s modl is about m. (a) Dtrmin t attractiv lctrical forc btwn an lctron and a proton sparatd by tis distanc. (b) Dtrmin (in V) t lctrical potntial nrgy of t atom. Givn: r = m Elctron and proton intract via t Coulomb forc qq 1 F = k = r 9 19 ( N m C )( C) 10 ( m) = N Find: Potntial nrgy is (a) F =? (b) PE =? qq 1V = = = r J 1 18 PE k.3 10 J 14 V

5 Difficultis wit t Rutrford Modl Atoms mit crtain discrt caractristic frquncis of lctromagntic radiation T Rutrford modl is unabl to xplain tis pnomna Rutrford s s lctrons ar undrgoing a cntriptal acclration and so sould radiat lctromagntic wavs of t sam frquncy, tus lading to lctron falling on a nuclus in about 10-1 sconds!!! 8. Emission Spctra A gas at low prssur as a voltag applid to it A gas mits ligt caractristic of t gas Wn t mittd ligt is analyzd wit a spctromtr, a sris of discrt brigt lins is obsrvd Eac lin as a diffrnt wavlngt and color Tis sris of lins is calld an mission spctrum T radius sould stadily dcras as tis radiation is givn offo T lctron sould vntually spiral into t nuclus It dosn t Emission Spctrum of Hydrogn Absorption Spctra T wavlngts of ydrogn s s spctral lins can b found from 1 = R λ 1 1 n R H is t Rydbrg constant R H = x 10 7 m -1 n is an intgr, n = 1,, 3, T spctral lins corrspond to diffrnt valus of n A.k.a. Balmr sris Exampls of spctral lins n = 3, λ = nm n = 4, λ = nm H An lmnt can also absorb ligt at spcific wavlngts An absorption spctrum can b obtaind by passing a continuous radiation spctrum troug a vapor of t gas T absorption spctrum consists of a sris of dark lins suprimposd on t otrwis continuous spctrum T dark lins of t absorption spctrum coincid wit t brigt lins of t mission spctrum

6 Applications of Absorption Spctrum 8.3 T Bor Tory of Hydrogn T continuous spctrum mittd by t Sun passs troug t coolr gass of t Sun s s atmospr T various absorption lins can b usd to idntify lmnts in t solar atmospr Ld to t discovry of lium (from lios) In 1913 Bor providd an xplanation of atomic spctra tat includs som faturs of t currntly accptd tory His modl includs bot classical and non-classical idas His modl includd an attmpt to xplain wy t atom was stabl 1 Bor s s Assumptions for Hydrogn Bor s s Assumptions T lctron movs in circular orbits around t proton undr t influnc of t Coulomb forc of attraction T Coulomb forc producs t cntriptal acclration Only crtain lctron orbits ar stabl Ts ar t orbits in wic t atom dos not mit nrgy in t form of lctromagntic radiation Trfor, t nrgy of t atom rmains constant and classical mcanics can b usd to dscrib t lctron s s motion Radiation is mittd by t atom wn t lctron jumps from a mor nrgtic initial stat to a lowr stat T jump cannot b tratd classically E E = f i f 3 Mor on t lctron s jump : T frquncy mittd in t jump is rlatd to t cang in t atom s s nrgy It is gnrally not t sam as t frquncy of t lctron s orbital motion T siz of t allowd lctron orbits is dtrmind by a condition imposd on t lctron s s orbital angular momntum mvr = n, n= 1,,3,... π E E = f i f 4 6

7 Rsults Bor Radius T total nrgy of t atom Nwton s s law 1 E = KE+ PE = m v k r v F = ma or k = m Tis can b usd to rwrit kintic nrgy as Tus, t nrgy can also b xprssd as r mv KE = k r r k E = r T radii of t Bor orbits ar quantizd ( = π ) n rn = n = 1,, 3, m k (from quantizd angular momntum: m vr = nħ and Nwtons law) Tis sows tat t lctron can only xist in crtain allowd orbits dtrmind by t intgr n Wn n = 1, t orbit as t smallst radius, calld t Bor radius,, a o a o = nm 5 6 Radii and Enrgy of Orbits A gnral xprssion for t radius of any orbit in a ydrogn atom is r n = n a o T nrgy of any orbit is E n = V/ n T lowst nrgy stat is calld t ground stat Tis corrsponds to n = 1 Enrgy is 13.6 V T nxt nrgy lvl as an nrgy of 3.40 V T nrgis can b compild in an nrgy lvl diagram T ionization nrgy is t nrgy ndd to compltly rmov t lctron from t atom (n =,, E = 0) T ionization nrgy for ydrogn is 13.6 V 7 Enrgy Lvl Diagram T valu of R H from Bor s s analysis is in xcllnt agrmnt wit t xprimntal valu A mor gnralizd quation can b usd to find t wavlngts of any spctral lins For t Balmr sris, n f = For t Lyman sris, n f = 1 Wnvr a transition occurs btwn a stat, n i and anotr stat, n f (wr n i > n f ), a poton is mittd T poton as a frquncy f = (E( i E f )/ and wavlngt λ 1 = R λ H 1 nf 1 ni 8 7

8 Problm: Transitions in t Bor s s modl A poton is mittd as a ydrogn atom undrgos a transition from t n = 6 stat to t n = stat. Calculat t nrgy and t wavlngt of t mittd d poton. A poton is mittd as a ydrogn atom undrgos a transition from t n = 6 stat to t n = stat. Calculat t nrgy and t wavlngt of t mittd poton. Givn: n i = 6 n f = From = R, H λ nf n i or Wit n i = 6 and n f = w av: 1 nn i f λ = R H ni n f 1 (36)(4) λ = 7 1 = = m m 410 nm Find: (α) λ =? (b) E γ =? T Poton nrgy is: 34 8 ( Jis)( m/s) 9 c E = λ = = = m J 3.03 V 9 30 Bor s s Corrspondnc Principl Succsss of t Bor Tory Bor s Corrspondnc Principl stats tat quantum mcanics is in agrmnt wit classical pysics wn t nrgy diffrncs btwn quantizd lvls ar vry small Similar to aving Nwtonian Mcanics b a spcial cas of rlativistic mcanics wn v << c Explaind svral faturs of t ydrogn spctrum Accounts for Balmr and otr sris Prdicts a valu for R H tat agrs wit t xprimntal valu Givs an xprssion for t radius of t atom Prdicts nrgy lvls of ydrogn Givs a modl of wat t atom looks lik and ow it bavs Can b xtndd to ydrogn-lik atoms Tos wit on lctron Z nds to b substitutd for in quations Z is t atomic numbr of t lmnt

9 Rcall Bor s s Assumptions Only crtain lctron orbits ar stabl. Radiation is mittd by t atom wn t lctron jumps from a mor nrgtic initial stat to a lowr stat E E = f i f T siz of t allowd lctron orbits is dtrmind by a condition imposd on t lctron s s orbital angular momntum Modifications of t Bor Tory Elliptical Orbits Sommrfld xtndd t rsults to includ lliptical orbits Rtaind t principl quantum numbr,, n Addd t orbital quantum numbr, l l rangs from 0 to n-1 n 1 in intgr stps All stats wit t sam principl quantum numbr ar said to form a sll T stats wit givn valus of n and l ar said to form a subsll mvr= n, n= 1,,3,... Wy is tat? Modifications of t Bor Tory Zman Effct and fin structur 8.5 d Brogli Wavs Anotr modification was ndd to account for t Zman ffct T Zman ffct is t splitting of spctral lins in a strong magntic fild Tis indicats tat t nrgy of an lctron is sligtly modifid wn t atom is immrsd in a magntic fild A nw quantum numbr, m l, calld t orbital magntic quantum numbr, ad to b introducd m l can vary from - l to + l in intgr stps On of Bor s s postulats was t angular momntum of t lctron is quantizd, but tr was no xplanation wy t rstriction occurrd d Brogli assumd tat t lctron orbit would b stabl only if it containd an intgral numbr of lctron wavlngts Hig rsolution spctromtrs sow tat spctral lins ar, in fact, f two vry closly spacd lins, vn in t absnc of a magntic fild Tis splitting is calld fin structur Anotr quantum numbr, m s, calld t spin magntic quantum numbr, was introducd to xplain t fin structur

10 d Brogli Wavs in t Hydrogn Atom QUICK QUIZ 1 In tis xampl, tr complt wavlngts ar containd in t circumfrnc of t orbit In gnral, t circumfrnc must qual som intgr numbr of wavlngts but π r = nλ, λ = 1,,3,... λ = mv, so mvr= n, n= 1,,3,... In an analysis rlating Bor's tory to t d Brogli wavlngt of lctrons, wn an lctron movs from t n = 1 lvl to t n = 3 lvl, t circumfrnc of its orbit bcoms 9 tims gratr. Tis occurs bcaus (a) tr ar 3 tims as many wavlngts in t nw orbit, (b) tr ar 3 tims as many wavlngts and ac wavlngt is 3 tims as long, (c) t wavlngt of t lctron bcoms 9 tims as long, or (d) t lctron is moving 9 tims as fast. (b). T circumfrnc of t orbit is n tims t d Brogli wavlngt (πr = nλ), so tr ar tr tims as many wavlngts in t n = 3 lvl as in t n = 1 lvl. Tis was t first convincing argumnt tat t wav natur of mattr was at t art of t bavior of atomic systms Quantum Mcanics and t Hydrogn Atom On of t first grat acivmnts of quantum mcanics was t solution of t wav quation for t ydrogn atom T significanc of quantum mcanics is tat t quantum numbrs and t rstrictions placd on tir valus aris dirctly from t matmatics and not from any assumptions mad to mak t tory agr wit xprimnts Problm: wavlngt of t lctron Dtrmin t wavlngt of an lctron in t tird xcitd orbit of t ydrogn atom, wit n =

11 Dtrmin t wavlngt of an lctron in t tird xcitd orbit of t ydrogn atom, wit n = 4. Quantum Numbr Summary Givn: Rcall tat d Brogli s wavlngt of lctron dpnds on its momntum, λ = /(m v). Lt us find it, n = 4 Rcall tat n mvr n = n, so mv = r rn n a0, so mv = = n π a n ( ) 0 Find: λ =? = = a0 n = nm = 1.33nm mv Tus, λ ( π ) π( ) T valus of n can incras from 1 in intgr stps T valus of l can rang from 0 to n-1 n 1 in intgr stps T valus of m l can rang from -l to l in intgr stps 41 4 QUICK QUIZ How many possibl orbital stats ar tr for (a) t n = 3 lvl of ydrogn? (b) t n = 4 lvl? T quantum numbrs associatd wit orbital stats ar n,, and m. For a spcifid valu of n, t allowd valus of rang from 0 to n 1. For ac valu of, tr ar ( + 1) possibl valus of m. (a) If n = 3, tn = 0, 1, or. T numbr of possibl orbital stats is tn [(0) + 1] + [(1) + 1] + [() + 1] = = 9. (b) If n = 4, on additional valu of is allowd ( = 3) so t numbr of possibl orbital stats is now 9 + [(3) + 1] = = 16 (ac is n ) Spin Magntic Quantum Numbr It is convnint to tink of t lctron as spinning on its axis T lctron is not pysically spinning Tr ar two dirctions for t spin Spin up, m s = ½ Spin down, m s = -½ Tr is a sligt nrgy diffrnc btwn t two spins and tis accounts for t Zman ffct Sodium D lins ar an xampl of tis splitting

12 Elctron Clouds Elctron Clouds T grap sows t solution to t wav quation for ydrogn in t ground stat T curv paks at t Bor radius T lctron is not confind to a particular orbital distanc from t nuclus T probability of finding t lctron at t Bor radius is a maximum T wav function for ydrogn in t ground stat is symmtric T lctron can b found in a sprical rgion surrounding t nuclus T rsult is intrprtd by viwing t lctron as a cloud surrounding t nuclus T dnsst rgions of t cloud rprsnt t igst probability for finding t lctron