ELEMENTARY QUANTUM MECHANICS

Transcription

1 Pag 1 of 9 ELEMENTARY QUANTUM MECHANICS 1. INTRODUCTION Th triumph of modrn physics is th triumph of quantum physics. Quantum physics (mchanics) Grw out of failurs of classical physics Launchd by xprimnts such as Photolctric ffct Compton scattring Blackbody radiation Found som solutions in Planck s hypothsis Wav-particl duality [D Brogli s hypothsis] Gnratd nw idas lik Wav function Schröndingr quation Hisnbrg s uncrtainty principl Quantum statistics. LIGHT AS A WAVE In classical lmntary physics light is considrd as an lctromagntic (EM) wav with a frquncy. y E y E 0 Dirction of Propagation x Vlocity c z x B z B 0 An lctromagntic wav is a travling wav with tim-varying lctric and magntic filds that ar prpndicular to ach othr and to th dirction of propagation. Elctric fild E y at position x at any tim t, wavnumbr propagation constant k ( x, t) E0 sin( kx ωt) Similar is th cas for B z. E y Vlocity (phas vlocity) of th wav, c ν, whr is th wavlngth 1 Intnsity of th light wav nrgy flowing pr unit ara pr scond, I cε 0E0 π angular frquncy ( ω πν ), whr ν frquncy absolut prmittivity

2 Pag of 9 Exprimnt 1: Young s Doubl-Slit Exprimnt P Constructiv intrfrnc Dstructiv intrfrnc Photographic plat showing Young s frings Young s frings [dark & bright frings] can only b xplaind by intrfrnc of light wavs. Two wavs from slits S 1 and S whn rach th photographic plat [P] in phas, thy intrfr constructivly to giv ris to bright fring. Rquirmnt for constructiv intrfrnc: S 1 P S P n Whn th two wavs ar out of phas by / thy intrfr dstructivly to giv dark fring. 1 Rquirmnt for dstructiv intrfrnc: S 1P SP n + Exprimnt : X-ray Diffraction [Crystallography] Atomic plans Pattrns (a) and (b) can only b xplaind by using wav concpts for light (c). Th path diffrnc of th two wavs [fig. (c)] is d sin θ, whr d is th sparation of th atomic plans. For constructiv intrfrnc this must b n. Othrwis, wavs will intrfr dstructivly to cancl ach othr. Thus, wavs rflctd from adjacnt atomic plans intrfr constitut a diffractd bam only whn, d sinθ n whr n 1,, 3, Bragg s law basis for idntifying and studying various crystal structurs (crystallography) Young s doubl-slit and X-ray diffraction xprimnts clarly show that light xhibits wav-lik proprtis.

3 Pag 3 of 9 3. FAILURE OF CLASSICAL PHYSICS Classical physics that considrs light as wav fails to xplain th rsults of som xprimnts. Th rsults can only b xplaind by introducing nw concpts that form th basis of quantum physics. 4. EXPERIMENT 1: THE PHOTOELECTRIC EFFECT Th dtails of th photolctric ffct wr in dirct contradiction to th xpctations of vry wll dvlopd classical physics. Th xplanation markd on of th major stps toward quantum thory. Rf.: Hyprphysics ( Explanation of photolctric ffct Th rmarkabl aspcts of th photolctric ffct whn it was first obsrvd wr: 1. Th lctrons wr mittd immdiatly - no tim lag!. Incrasing th intnsity of th light incrasd th numbr of photolctrons, but not thir maximum kintic nrgy! 1 [Rcall I cε 0E0 ] 3. Rd light will not caus th jction of lctrons, no mattr what th intnsity! 4. A wak violt light will jct only a fw lctrons, but thir maximum kintic nrgis ar gratr than thos for intns light of longr wavlngths! Analysis of data from th photolctric xprimnt showd that th nrgy of th jctd lctrons was proportional to th frquncy of th illuminating light. This showd that whatvr was knocking th lctrons out had an nrgy proportional to light frquncy. Th rmarkabl fact that th jction nrgy was indpndnt of th total nrgy of illumination (light intnsity) showd that th intraction must b lik that of a particl which gav all of its nrgy to th lctron (on collision)! This fit in wll with Planck's hypothsis that light in th blackbody radiation xprimnt could xist only in discrt bundls with nrgy E hν, whr h is th Planck s constant (h 6.66X10-34 Js) Rf.: Hyprphysics (

4 Pag 4 of 9 Concpt of Work Function In th photolctric ffct xprimnt, lctrons jctd from a sodium mtal surfac wr masurd as an lctric currnt. Finding th opposing voltag it took to stop all th lctrons gav a masur of th maximum kintic nrgy of th lctrons in lctron volts. 3 Maximum photolctron kintic nrgy KE m in V 1 Slop ΔE /Δν 4.1X10-15 V s h Light blow a frquncy of 4.39X10 14 Hz or wavlngth longr than 683 nm would not jct lctrons Δν 3X10 14 Hz ΔE 1.5 V 4 ν X10 Frquncy, Hz 14 Data from Millikan, 1916 Th minimum nrgy rquird to jct an lctron from th surfac is calld th photolctric work function. Th thrshold for this lmnt corrsponds to a wavlngth 0 of 683 nm for sodium. Th slop of KE m vs ν givs h, th Planck s constant ( 4.1X10-15 V s 6.66X10-34 Js). Using th wavlngth 0 in th Planck rlationship, E hν 0, (c ν 0 ) givs a photon nrgy of 1.8 V. Einstin in 1905 succssfully xplaind th photolctric ffct. Light consists of nrgy packts or photons, ach with nrgy hν. An lctron in a mtal is in a lowr stat of potntial nrgy (PE) than in vacuum, by an amount Φ, th work function of th mtal. This lowr PE is a rsult of th Coulombic attraction intraction btwn th lctron and th positiv mtal ions. Upon collision som of th photon nrgy gos toward ovrcoming th PE barrir. Th nrgy lft (hν Φ) givs th lctron its KE. Photomission only occurs whn hν is gratr than Φ. At thrshold, hν 0 Φ, whr ν 0 is th critical (cut-off) frquncy, blow which no photomission occurs.

5 Pag 5 of 9 Th Nobl Priz in Physics 191 "for his srvics to Thortical Physics, and spcially for his discovry of th law of th photolctric ffct" Albrt Einstin (Grmany and Switzrland) Kaisr-Wilhlm-Institut (now Max-Planck-Institut) für Physik Brlin, Grmany, b (in Ulm, Grmany), d Problm What is th nrgy of a blu photon that has a wavlngth of 450 nm? hc Solution: W know th photon nrgy, E ph hν J s 3 10 m s Thrfor, E blu J m Usually w xprss th photon nrgy in units of V. [1 V 1.6 X J] Thus J E blu. 75 V J V Problm In th photolctric xprimnt, grn light with wavlngth of 5 nm is th longst wavlngth radiation that can caus th photomission from a clan Sodium surfac. a. What is th work function of Na? hc W know work function, Φ h ν 0 Thrfor work function of Sodium, Φ J s 3 10 m m s J V 8 1 Na V b. If UV (ultraviolt) radiation of wavlngth 50 nm is incidnt to th Sodium surfac, what would b th kintic nrgy of th photomittd lctrons? Th xcss nrgy of th incidnt photon aftr ovrcoming th work function, i.., E ph Φ givs th photomittd lctron its K.E. 8 1 hc Js 3 10 ms Th nrgy of th incidnt photon, E UV J m J Or, E UV V J V Thrfor, K.E V.38 V.58 V

6 Pag 6 of 9 5. EXPERIMENT : THE COMPTON SCATTERING Arthur Holly Compton obsrvd scattring of x-rays from lctrons in a carbon targt. Whn an X-ray with frquncyν, wavlngth striks an lctron it is dflctd or scattrd lctron rcoils and movs away scattrd X-ray has a frquncy ν which is lss than ν y X -ray ph oto n ν, x c Elctro n Rcoilinglctron φ θ Scattrd photon It is found that th rcoiling lctrons possss kintic nrgy, K.E. hν - hν Sinc th lctron now also has a momntum p, from law of consrvation of momntum (momntum bfor collision momntum aftr collision), w conclud that X-ray also has a momntum. h Momntum of th photon is rlatd to its wavlngth by p. Th Nobl Priz in Physics 197 "for his discovry of th ffct namd aftr him" Arthur Holly Compton (USA) Univrsity of Chicago Chicago, IL, USA b. 189 d. 196 h ν', ' Δ ( 1 cosθ ) m c c Problm Calculat th nrgy and momntum of an X-ray photon with a wavlngth of 0.6 angstrom and th vlocity of a corrsponding lctron that has th sam momntum. Solution 1 hc Js 3 10 ms Enrgy of th X-ray photon, hν m J V h Momntum of th X-ray photon, p s m Corrsponding lctron with th momntum 8 4 E ph Js kg m 10 p m v will hav a vlocity, kg m s p v ms m kg V

7 Pag 7 of 9 6. EXPERIMENT 3: BLACK BODY RADIATION All objcts mit and absorb nrgy in th form of radiation (thrmal radiation). Intnsity of th radiation dpnds on th radiation wavlngth and tmpratur of th objct. At thrmal quilibrium (objct and surroundings at th sam tmpratur), Enrgy absorbd Enrgy mittd Whn objct tmp. > surrounding tmp. Thr is a nt mission of radiation nrgy. Black body radiation maximum amount of radiation nrgy that can b mittd by an objct In gnral intnsity of th radiatd nrgy dpnds on th matrial s surfac But radiation mittd from a cavity [box] with a small aprtur indpndnt of th matrial of th cavity and corrsponds vry closly to black body radiation I c a n i a ir d al ctr p S 3000 K 500 K Clas s ica lth ory Pl an c k's rad i at io nlaw ( μ m) Schmatic illustration of black body radiation and its charactristics (spctral irradianc I is th mittd radiation intnsity (powr pr unit ara) pr unit wavlngth) Classical physics xplanation acclration, dclration of th chargs du to various thrmal vibrations, oscillations, or motions of th atoms in th surfac rgion of th cavity matrial rsult in lctromagntic wavs of mission ths EM wavs intrfr with ach othr giving ris to many typs of standing EM wavs (mods) with diffrnt wavlngths in th cavity radiation mod ach mod contributs an nrgy kt to th mittd intnsity classical Rayligh-Jans law prdicts 1 Spctral irradianc, I 4 and I T but th law prdicts continud I incras at lowr Wavlngth lading to ultraviolt catastroph this is not in agrmnt with th xprimnt mods dvlopd in cavity Classical physics faild to xplain th black body radiation xprimntal rsults!

8 Pag 8 of 9 Max Planck s xplanation Max Planck xplaind th black body radiation xprimntal rsults in radiation within th cavity involvs th mission and absorption of discrt (sparat, isolatd, distinct) amounts of light nrgy by th mods dvlopd in th cavity th quantity of th discrt nrgy quantum is dtrmind by th frquncy ν of th radiation and givn by hν mods mit and absorb an intgr multipl of th discrt nrgy quantum, i.. nhν probability of a mod to hav nhν nrgy is proportional to th Boltzmann factor nhν kt πhc spctral irradianc is givn by, I Planck s black body hν 5 kt radiation formula 1 th formula is in xcllnt agrmnt with all obsrvd black body radiation charactristics Th Nobl Priz in Physics 1918 "in rcognition of th srvics h rndrd to th advancmnt of Physics by his discovry of nrgy quanta" Max Karl Ernst Ludwig Planck (Grmany) Brlin Univrsity, Brlin, Grmany b. 1858, d ELECTRON AS WAVE (DE BROGLIE RELATIONSHIP) So far w hav sn light can bhav as if it wr a stram of particl-lik ntitis calld photons. Can lctrons (which ar particls) xhibit wav-lik proprtis (intrfrnc, diffraction)? To prob Young s doubl-slit and Crystallographic xprimnts ar carrid out with nrgtic lctron bams. Th Young s doubl-slit xprimnt showd Young s frings (high an low-intnsity rgions) Th Crystallographic xprimnt producd diffraction rings. Indd lctrons xhibit wav-lik proprtis! Th lctron bams oby Braggs diffraction condition, d sinθ n, knowing th intratomic distanc d and masuring th angl of diffraction θ w can valuat th wavlngth associatd with th wav-lik bhaviour of th lctrons. Th momntum of th lctrons can b dtrmind from th fact that th kintic nrgy gaind by th lctrons p / m is qual to th nrgy V from th acclrating voltag V in th lctron tub.

9 Pag 9 of 9 It is found that an lctron travling with a momntum p bhavs lik a wav of wavlngth h givn by p Earlir from th Compton scattring w found that th photon with wavlngth bhavs h lik a particl of momntum p givn by p h h Thus or p rlats wav-lik and particl-lik proprtis to and from ach p othr (wav-particl duality). This is first hypothsizd by d Brogli in 194 known as d Brogli rlations Th Nobl Priz in Physics 199 "for his discovry of th wav natur of lctrons" Princ Louis-Victor Pirr Raymond d Brogli (Franc) Sorbonn Univrsity, Institut Hnri Poincaré, Paris, Franc b. 189, d. 1987