Chapter 40 Introduction to Quantum

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Baldric Lyons
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Catr 0 Introdution to Quantu Pysis 900-90 A nw tory alld quantu anis was igly sussful in xlaining t bavior of artils of irosoi siz.

In addition, ty bliv tat t wav natur sall b nand for artils of irosoi siz just baus of t orn of tir wavs in su a sall sal. 0.

1 Catr 0 Introdution to Quantu Pysis A nw tory alld quantu anis was igly sussful in xlaining t bavior of artils of irosoi siz. Baus sintists larn t wav and artil naturs of ligt in 9 t ntury, ty roos t sa dual naturs to artils. In addition, ty bliv tat t wav natur sall b nand for artils of irosoi siz just baus of t orn of tir wavs in su a sall sal. 0. Blakbody Radiation and Plank s Hyotsis. Total owr of t ittd radiation inras wit tratur. St s Law: P AT. T ak of t wavlngt distribution sifts to sortr wavlngt as t tratur inrass. Win s dislant law: axt.8980 K Classial Aroa: S / k S / k k ln N, V PN, V, or PN, V, C S,

2 Fro d dq dw dn S T PV N Canonial nsbl: TdS PdV dn, w av P / N, V, T C (or lassial statistis, Boltzann distribution) T Boltzann distribution: T robability of finding a artil at nrgy is P x. R Cn n R C' n n rinil of irosoi rvrsibility: C C' n n n n lnn lnn lnn n ln lasti sattring: T only way to satisfy t two qs is ln n A n Ax A 0 Noraliz to gt t onstant A by d A P P d us intgration P d ln d d d T robability of finding a artil wit a sd btwn v and v+dv is N v N( ) / v v / ltroagnti wavs ar onfind in a D box wit a lngt of L. x, t Asin kx t Asin kx t Asin kx os t and kl n k n n L For a D box: k / L k N / L k 8 k k / L Considr two ossibl olarizations, t nubr of stats sall b:

3 k dk k k dk B / L k, T dk dk k T V k, T dk, T d k Td 8 ang t dirtion of t intgration I kbt, T, T (Raylig-Jans Law, SI Gaussian unit) B T L B Quantu Aroa (Max Plank):. T nrgy of an osillator an av only rtain disrt valus n nf. n is a ositiv intgr alld quantu nubr. T nrgy is quantizd. a disrt valu orrsonds to a diffrnt quantu stat.. T aount of nrgy ittd by t osillator and arrid by t quantu of radiation is f. Plank s onst: J s P nf us suation n f n f nf nf nf nf / ln f f / n T n n f f f / 8 f T d dk T I, f / T, T 5, / f f f / f / (lassial liit) 8 5, / xal: (a) Find t ak wavlngt of t blakbody radiation ittd by t uan body wn t skin tratur is 5 o C. T.898 K ax 9. Infrard ligt ax 0

4 (b) Find t ak wavlngt of t blakbody radiation ittd by t tungstn filant of a ligtbulb wi orats at 000 K.. ax () Find t ak wavlngt of t blakbody radiation ittd by t sun (5800 K). ax 0. 5 blu ligt xal: T Quantizd Osillator A.0 kg blok is attad to a asslss sring tat as a for onstant of k = 5 N/. T sring is strtd 0.0 fro its quilibriu and rlasd fro rst. (a) Find t total nrgy of t syst and t frquny of osillation aording to lassial alulations. kx 50. J k 5 f 0.56 Hz (b) Assu t nrgy of t osillator is quantizd, find t quantu nubr n for t syst osillating wit tis alitud. f n n T Potoltri fft nrgy of Potons: f, J s Rquird nrgy to ov an ltron out of t tal Work funtion vauu K ax v f V s, Vs is t stoing otntial.

5 Rquird nrgy for an ltron to b rovd fro t tal: f 0 f 0 --> t f t xal: Calulat t oton nrgis for ligt of wavlngts 00 n (violt) and 700 n (rd). f xal: T intnsity of sunligt at t art s surfa is aroxiatly 00 W/. Assu t avrag oton nrgy is V (orrsonding to a wavlngt of aroxiatly 600 n), alulat t nubr of otons tat strik an ara of a sond. W 00 ( s) , N T Coton fft T sattring of otons fro argd artils is alld Coton sattring aftr Artur Coton wo was t first to asur oton-ltron sattring in 9. Wn t inoing oton givs art of its nrgy to t ltron, tn t sattrd oton as lowr nrgy and aording to t Plank rlationsi as lowr frquny and longr wavlngt. T wavlngt ang in su sattring dnds only uon t angl of sattring for a givn targt artil. Wy is t frquny of sattrd ligt diffrnt fro tat of inidnt ligt? T iortan of 5

6 For otons: f Montu Consrvation: os 0 nrgy of t sattrd ltron: Rst nrgy: 0, Kinti nrgy: K, Total nrgy: rst K nrgy Consrvation: oton, ltron oton, ltron --> os os --> os --> os C. --> Can w us visibl ligt to rfor t xrints of Coton sattring? v 0. 9., 0.9 K f 6

7 xal: T X-ray oton of wavlngt 6 aks a ad-on ollision wit an ltron, so tat t sattrd oton gos in a dirtion oosit to tat of t inidnt oton. T ltron is initially at rst. (a) How u longr is t wavlngt of t sattrd oton tan tat of t inidnt oton? (b) Wat is t kinti nrgy of t roiling ltron? os. 86 C C K Vn 9500 V 0. Potons and ltroagnti Wavs T otoltri fft and t Coton fft rovid tat t ligt bav as if it was a artil. T ontu of t ligt artil (oton) is P=/= /. Wi odl is orrt? Is ligt a wav or a artil? T answr dnds on t nona bing obsrvd. W ust at bot odls and adit tat t tru natur of ligt is not dsribabl in trs of any singl lassial itur. T artil odl and t wav odl of ligt olnt a otr. Partil Natur:, Wav Natur: f, Tir Corrsonding Rlationsi: f f k 7

0.5 T Wav Prortis of Partils Baus otons av bot wav and artil aratristis, ras

For a irosoi artil wit, P : T wavlngt of t oton wav: /.

an b dsribd by t wav xrssion Asin kx t Asin x t.

8 0.5 T Wav Prortis of Partils Baus otons av bot wav and artil aratristis, ras all fors of attr av bot rortis. For a irosoi artil wit, P : T wavlngt of t oton wav: /. T frquny of t ltron wav: f, t frquny of t artil is dndnt on its nrgy T artil an b dsribd by t wav xrssion Asin kx t Asin x t. 9 d Brogli s attr wav 96 - Davisson and Grr sudd in asuring t wavlngt of ltrons. T Davisson-Grr xrint Tir rsults sow onlusivly t wav natur of ltrons and onfird t d Brogli rlationsi /. xal: Find t d Brogli wavlngt of a 6 0 /s g artil oving wit a sd of xal: Calulat t d Brogli wavlngt for an ltron ( 9.0 kg) 8

oving at.00 X 0 7 /s. / / 6.660 9.0 0 v 7.80 7 For a low nrgy ltron wit kinti nrgy K, its ontu is found fro K --> K.6 T wavlngt is. --> n ( K in ltron volts) K K 0.

9 oving at.00 X 0 7 /s. / / v For a low nrgy ltron wit kinti nrgy K, its ontu is found fro K --> K.6 T wavlngt is. --> n ( K in ltron volts) K K 0.6 T Quantu Partil T quantu artil is a obination of t artil odl (Catr ) and t wav odl (Catr 6). idal artil: loalizd in sa idal wav: singl frquny, infinitly long, dloalizd in sa If a larg nubr of wavs ar obind, t rsult is a wav akt wi rrsnts a artil. Fro wav akt to ontinuous wav? artil to wav Fro ontinuous wav to wav akt? wav to artil bats 9

y os k k k Aos x t os x - as vloity k t d - grou vloity artil vloity k dk For

10 wav akt Wat ar t wav vloity and artil vloity of t ligt? wav vloity as vloity artil vloity grou vloity A k x t, A k x t y os y v as v g y os k k k Aos x t os x - as vloity k t d - grou vloity artil vloity k dk For ligt: v grou d d dk d k k For quantu artil: v grou d dk d d k d d d d u 0.7 T Doubl-Slit xrint Rvisitd 0

11 0.8 T Unrtainty Prinil xtnsion of wav artil duality ont: artil loalizd in sa wav dloalizd in sa wat about t dtrination of bot wav and artil natur? x k 0 x k x 0 Asin kx t Asink k x t k k k k Asin x t os x Aos T Natur of intrfrn wit itslf: xsinkx t ltron ovs, Asin x t asur intrfrn wit itslf to giv you, inforation ang of aus k & k kx k x / T unrtaind osition of tis wav akt is k x x x, k x T satial distribution of t oton: T ontu arrid by t oton is f f T rodut of t intrinsi unrtaintis in osition and ontu is: x ~

12 T unrtainty rinil: x x x vt v t t instin s Pitur: ltron ltron xal: Loating an ltron T sd of an ltron is asurd to b 5.00 X 0 /s to an auray of %. Find t iniu unrtainty in dtrining t osition of tis ltron. x

CHEM 333 QUANTUM THEORY AND SPECTROSCOPY PROBLEM SET I SOLUTION KEY

CHEM 333 QUANTUM THEORY AND SPECTROSCOPY PROBLEM SET I SOLUTION KEY CHEM QUANTUM THEORY AND SPECTROSCOPY PROLEM SET I SOLUTION KEY. T for onstant of O is N. -. Assuing a aroni osillator odl for t vibrational otion of O, wat is t diatoi vibrational frquny in - units? v