PHYS-3301 Lecture 9. CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I. 5.3: Electron Scattering. Bohr s Quantization Condition
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1 CHAPTER 5 Wave Properties of Matter ad Quatum Mecaics I PHYS-3301 Lecture 9 Sep. 5, X-Ray Scatterig 5. De Broglie Waves 5.3 Electro Scatterig 5.4 Wave Motio 5.5 Waves or Particles? 5.6 Ucertaity Priciple 5.7 Probability, Wave Fuctios, ad te Copeage Iterpretatio 5.8 Particle i a Box Bor s Quatizatio Coditio Oe of Bor s assumptios cocerig is ydroge atom model was tat te agular mometum of te electro-ucleus system i a statioary state is a itegral multiple of /π. (see below) Te electro is a stadig wave i a orbit aroud te proto. Tis stadig wave will ave odes ad be a itegral umber of wavelegts. Te agular mometum becomes: 5.3: Electro Scatterig Davisso ad Germer experimetally observed tat electros were diffracted muc like x rays i ickel crystals. Davisso-Germer experimet Electros are produced by te ot filamet, accelerated, ad focused oto te target. Electros are scattered at a agle ϕ ito a detector, wic is movable. Te distributio of electros is measured as a fuctio of ϕ. Te etire apparatus is located i a vacuum. D-G data for scatterig of electros from Ni. Te peak ϕ = 50 builds dramatically as te eergy of te electro ears 54 ev. A scematic diagram of stadig waves i a electro orbit aroud a ucleus. A itegral umber of wavelegts fits i te orbit. Note tat te electro does ot wiggle aroud te ucleus. Te displacemet from te dased lie represets its wave amplitude.
2 5.3: Electro Scatterig Davisso ad Germer experimetally observed tat electros were diffracted muc like x rays i ickel crystals. D = iteratomic spacig d = lattice plae spacig 5.3: Electro Scatterig Determie te De Broglie s wavelegt for a 54-eV scattered electro. l = c = 140eV m = 0.167m Kmc 5 54eV ev ( ) λ = d si θ ad λ = D si Φ Explaatio: costructive iterferece of electro waves reflected from atomic plaes i te crystal. - æ l ö l = dcosa f = a = cos ç = 51 èd ø cosa = l/d Wave Motio: Properties of Matter Waves Wat properties caracterize a wave? l, f, v, also A tat varies wit positio/time (i.e. (x,) Wave fuctio: we use differet symbol for wave fuctio of differet kids of wave (1) For trasverse wave o strig; y(x, -- strig s trasverse displacemet () For E&M plae wave; E(x, &B(x, -- describe ow te oscillatig E- & B-field vary wit (x, (3) For matter wave; y(x, = probability amplitude y(x, è tell us probability of fidig te particle (see ext slide for some details; also 5.7 i details) Amplitude = fuctio of (x, Properties of Matter Waves! ( x, Probability Amplitude
3 Amplitude = fuctio of (x, Te probability desity to fid a particle at coordiate x, at time t! ( x, Probability Amplitude! ( x, Te probability to fid a particle i a iterval "x, "t P dxdt! ( x, =!! Itegrate over "x, "t x 4. Properties of Matter Waves Wave legt p l = = De Broglie s ypotesis: k p ç=è l = Wavelegt of matter wave Experimetally cofirmed; e.g. x-tal diffractio de Broglie got Nobel prize i 199 l = C mc t P ( x) dt! ( x, =! Frequecy (of matter waves) f = E/ ow, it s ope more coveiet to express l = /p ad f = E/ i terms of te followig quatities; k = p/l (wave umber), w = p/t (agular frequecy) Aoter coveiet defiitio: = /p = x Js express te fudametal wave-particle relatiosip as! p = "! k E =!ω
4 Wavelegt Frequecy Properties of Matter Waves De Broglie (199)! = f = p E Properties of Matter Waves Speed v = f! It does give te wave speed v = (E/)(/p) = E/p c.f. for poto, E = pc BUT v is NOT te velocity of te massive particle, it s te velocity of te Matter wave Electromagetic waves (lig Particles (potos) Waves? Massive Particles
5 Te Free-Particle Scrodiger Wave Equatio Aalogy: [Q] How do we determie te y(x, of a matter wave? Waves o a strig wave speed Waves o a strig wave speed Solutio = fuctio Plae Wave E&M Wave E&M Wave
6 5.4: Wave Motio Wave Properties De Broglie matter waves suggest a furter descriptio. Te displacemet of a wave is Te pase velocity is te velocity of a poit o te wave tat as a give pase (for example, te cres ad is give by Tis is a solutio to te wave equatio A pase costat Φ sifts te wave:. Defie te wave umber k ad te agular frequecy ω as: ad Te wave fuctio is ow: Ψ(x, = A si (kx ω Priciple of Superpositio We two or more waves traverse te same regio, tey act idepedetly of eac oter. Combiig two waves yields: Priciple of Superpositio We two or more waves traverse te same regio, tey act idepedetly of eac oter. Combiig two waves yields: Te combied wave oscillates witi a evelope tat deotes te maximum displacemet of te combied waves. We combiig may waves wit differet amplitudes ad frequecies, a pulse, or wave packet, is formed wic moves at a group velocity:
7 Fourier Series Wave Packet Evelope Te sum of may waves tat form a wave packet is called a Fourier series: Te superpositio of two waves yields a wave umber (k) ad agular frequecy (w) of te wave packet evelope. Summig a ifiite umber of waves yields te Fourier itegral: We ca idetify a localized regio Dx = x x 1 were x 1 & x represet two cosecutive poits were te evelope is zero. Tis must be differet by a pase of p for te values x 1 & x, because x x 1 represet oly oe alf of te wavelegt of te evelope cofiig te wave. Te rage of wave umbers ad agular frequecies tat produce te wave packet ave te followig relatios: (1/) Dk x - (1/) Dk x 1 = p Similarly, Wave Packet Evelope Te superpositio of two waves yields a wave umber (k) ad agular frequecy (w) of te wave packet evelope. Gaussia Fuctio Gaussia wave packets are ofte used to represet te positio of particles, because te associated itegrals are relatively easy to evaluate. A Gaussia wave packet describes te evelope of a pulse wave. Te rage of wave umbers ad agular frequecies tat produce te wave packet ave te followig relatios: A Gaussia wave packet as similar relatios: Te localizatio of te wave packet over a small regio to describe a particle requires a large rage of wave umbers. Coversely, a small rage of wave umbers caot produce a wave packet localized witi a small distace. Te group velocity is
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