Chapter 37 The Quantum Revolution

Transcription

1 Chaptr 37 Th Quantum Rvolution Max Plank Th Nobl Priz in Physis 1918 "in rognition of th srvis h rndrd to th advanmnt of Physis by his disovry of nrgy quanta" Albrt Einstin Th Nobl Priz in Physis 191 "for his srvis to Thortial Physis, and spially for his disovry of th law of th photoltri fft" Th mystry of partils and wavs

2 Blakbody Radiation th lassial pitur Th lassial radiation fild: u f ( T ) 8πf 3 ε In a lassial statistial thory th avrag nrgy pr dgr of frdom is: ε k B T To th Rayligh-Jans law for a blak body mittr u ( T ) 8πf f 3 k B T Th ultraviolt atastroph

3 Blakbody Radiation toward th Quantum Hypothsis Dfin gomtrial sris: Z( x) nx 1+ d x Z( x) x dx x d dx + x nx Plank: nrgy of th osillating mods om in disrt portions ε nhf Probability that ε ours in th nrgy distribution of th avity (Maxwll Boltzmann) ε / kt p( ε ) Man nrgy εp ε p With: 1 + L 1 x n nx x ( ε ) ( ε ) x ε n p dε n dε p hf kt ktx ε Z( x) n d dx ( ε ) n nhf n nhf / kt n n ktx ( ε ) nhf / kt n n Z( x) ktx x ε ktx 1 x d dx ln Z( x) ktx ktx x 1 n d dx hf hf / kt nx nx ln(1 1 x )

4 Plank s Quantum Hypothsis; Blakbody Radiation 1 1 ) ( 4 ) ( / 3 kt hf f f hf T u T I π ) ( / kt hf f hf f T u π ε π Radiation dnsity Radiation intnsity 1 1 ) ( ) ( / 5 kt h f h T I T I π Saling from frquny to wavlngth?

5 Plank s Quantum Hypothsis; Blakbody Radiation Plank found th valu of his onstant by fitting blakbody urvs to th formula giving Plank s proposal was that th nrgy of an osillation had to b an intgral multipl of hf. This is alld th quantization of nrgy.

6 Drivation of Win s law Radiation intnsity (Plank) I ( T ) I f ( T ) πh 5 1 h / kt 1 Dfin (dimnsionlss) I ( T ) π h ( kt ) 4 Univrsal shap: 3 5 x x 5 1 x g( x) h kt x x 5 1 dg x x Maximum: 5 0 dx x x For 1 x x /5 so xˆ max T h xk ˆ

7 Plank s Quantum Hypothsis; lading to Win s law Blakbody radiation for thr diffrnt tmpraturs Not that frquny inrass to th lft. Th rlationship btwn th tmpratur and pak wavlngth is givn by Win s law:

8 Plank s Quantum Hypothsis; lading to Stfan-Boltzmann s law Radiation intnsity I f ( T ) u 4 f ( T ) πhf 3 hf 1 / kt 1 Total intnsity I( T ) 0 πhf 3 hf df / kt 1 Us again: h x kt Thn I( T ) πh kt h 4 x 0 x 3 dx 1 Chk Mathmatia (or solv): 0 x x 3 4 dx π 1 15 Stfan- Boltzmann I( T ) σt π k With: 8 4 σ W/m K 3 15h

9 Photon Thory of Light and th Photoltri Efft Einstin suggstd that, givn th suss of Plank s thory, light must b mittd in small nrgy pakts: Ths tiny pakts, or partils, ar alld photons.. Einstin mad a stp furthr than th assumptions of Plank who doubtd th rality of th quanta

10 Photon Thory of Light and th Photoltri Efft Th photoltri fft: if light striks a mtal, ltrons ar mittd. Masurmnt of kinti nrgy of ltrons: Stopping potntial K max V 0 Masurmnts at varying f

11 Photon Thory of Light and th Photoltri Efft Th partil thory assums that an ltron absorbs a singl photon. Plotting th kinti nrgy vs. frquny: hf K + W W 0 is matrial proprty In som ass svral kinti nrgis masurd: Last bound ltrons orrspond to th work funtion: W 0 Minimum amount of nrgy rquird to rlas ltron hf K max + W 0 This shows lar agrmnt with th photon thory, and not with wav thory: No ltrons mittd for f < f 0

12 Photon Thory of Light and th Photoltri Efft If light is a wav, thory prdits: 1. Numbr of ltrons and thir nrgy should inras with intnsity.. Frquny would not mattr. If light is partils, thory prdits: Inrasing intnsity inrass numbr of ltrons but not nrgy. Abov a minimum nrgy rquird to brak atomi bond, kinti nrgy will inras linarly with frquny. Thr is a utoff frquny blow whih no ltrons will b mittd, rgardlss of intnsity. Conlusion: light onsists of partils with nrgy Ehf : photons

13 Enrgy, Mass, and Momntum of a Photon Clarly, a photon must travl at th spd of light. Looking at th rlativisti quation for momntum, it is lar that this an only happn if its rst mass is zro. p / mv / 1 v E p + m 4 W alrady know that th nrgy is hf; w an put this in th rlativisti nrgy-momntum rlation and find th momntum: A photon must hav dirtdnss (and momntum) as follows from th Compton fft

14 Compton Efft Compton xprimnts (193) sattrd X-rays from diffrnt matrials hav slightly longr wavlngth than th inidnt ons th wavlngth dpnds on th sattring angl: Arthur Compton Th Nobl Priz in Physis 197 "for his disovry of th fft namd aftr him"

15 Compton Efft This is anothr fft that is orrtly prditd by th photon modl and not by th wav modl. Bfor ollision photon ltron E hf E m h p h Aftr ollision photon ltron E ' E tot h ' γm h p ' ' E p kin γm v ( 1) m γ

16 Compton Efft Consrvation of nrgy Consrvation of momntum Along x: h h + γ ' h ( 1) m h osφ + γmv osθ ' Along y: h 0 sinφ γmvsinθ ' Thr quations with 3 unknowns, liminat v and θ Compton sattring: ' + h m ( 1 osφ)

17 Compton Efft Δ m h ( 1 osφ) ( 1 osφ) C Not that C ~ nm So th ffts is not so wll visibl with visibl light Compton prformd his xprimnt with x-rays

18 Photon Intrations; Pair Prodution Photons passing through mattr an undrgo th following intrations: 1. Photoltri fft: photon is ompltly absorbd, ltron is jtd.. Photon may b totally absorbd by ltron, but not hav nough nrgy to jt it; th ltron movs into an xitd stat. 3. Th photon an sattr from an atom and los som nrgy. 4. Th photon an produ an ltron positron pair. Minimum nrgy: E h m

19 Wav Natur of Mattr Just as light somtims bhavs lik a partil, mattr somtims bhavs lik a wav. Th wavlngth of a partil of mattr is. D Brogli wavlngth of mattr Louis D Brogli Th Nobl Priz in Physis 190 "for his disovry of th wav natur of ltrons"

20 Wav-Partil Duality; th Prinipl of Complmntarity W hav phnomna suh as diffration and intrfrn that show that light is a wav, and phnomna suh as th photoltri fft and th Compton fft that show that it is a partil. Whih is it? This qustion has no answr; w must apt th dual wav partil natur of light. Th prinipl of omplmntarity stats that both th wav and partil aspts of light ar fundamntal to its natur.

21 Eltron Mirosops Eltrons wavs usd for imaging Wavlngths of about nm. Transmission ltron mirosop th ltrons ar fousd by magnti oils Sanning ltron mirosop th ltron bam is sannd bak and forth aross th objt to b imagd.