PHYS-3301 Lecture 10. Wave Packet Envelope Wave Properties of Matter and Quantum Mechanics I CHAPTER 5. Announcement. Sep.
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1 Aoucemet Course webpage PHYS-3301 Lecture 10 HW3 (due 10/4) Chapter 5 4, 8, 11, 15, 22, 27, 36, 40, 42 Sep. 27, 2018 Exam 1 (10/4) Chapters 3, 4, & 5 CHAPTER 5 Wave Packet Evelope Wave Properties of Matter ad Quatum Mechaics I X-Ray Scatterig De Broglie Waves Electro Scatterig Wave Motio Waves or Particles? Ucertaity Priciple Probability, Wave Fuctios, ad the Copehage Iterpretatio 5.8 Particle i a Box The superpositio of two waves yields a wave umber (k) ad agular frequecy (w) of the wave packet evelope. The rage of wave umbers ad agular frequecies that produce the wave packet have the followig relatios: A Gaussia wave packet has similar relatios: The localizatio of the 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 withi a small distace.
2 Gaussia Fuctio Gaussia wave packets are ofte used to represet the positio of particles, because the associated itegrals are relatively easy to evaluate. A Gaussia wave packet describes the evelope of a pulse wave. Dispersio Cosiderig the group velocity of a de Broglie wave packet yields: The group velocity is Dispersio The relatioship betwee the phase velocity ad the group velocity is 5.5 Is It a Wave or a Particles? Duality E&M-Waves behavig like Particles v.s. Particles behavig like Waves! > D Wave Hece the group velocity may be greater or less tha the phase velocity. A medium is called odispersive whe the phase velocity is the same for all frequecies ad equal to the group velocity.! << D Particle
3 5.5. Is It a Wave or a Particles? Duality EM-Waves behavig like Particles v.s. Particles behavig like Waves 5.5: Waves or Particles? Youg s double-slit diffractio experimet demostrates the wave property of light. However, dimmig the light results i sigle flashes o the scree represetative of particles.! > D Wave! << D Particle Double-slit Diffractio Experimet light
4 INDIVIDUAL PHOTON HITS Although diffractio of light is a wave pheomeo, there is o smooth distributio of light i the diffractio patter, but the patter is rather formed of may idividual hits of particles the photos
5 A sigle photo DOES NOT get disitegrated i the Diffractio process to make a smooth diffractio patter Comig back to that soo Wave particle duality solutio 5.6: Ucertaity Priciple The solutio to the wave particle duality of a evet is give by the followig priciple. It is impossible to measure simultaeously, with o ucertaity, the precise values of k ad x for the same particle. The wave umber k may be rewritte as Bohr s priciple of complemetarity: It is ot possible to describe physical observables simultaeously i terms of both particles ad waves. For the case of a Gaussia wave packet we have Physical observables are those quatities such as positio, velocity, mometum, ad eergy that ca be experimetally measured. I ay give istace we must use either the particle descriptio or the wave descriptio. Thus for a sigle particle we have Heiseberg s ucertaity priciple:
6 Eergy Ucertaity If we are ucertai as to the exact positio of a particle, for example a electro somewhere iside a atom, the particle ca t have zero kietic eergy. The eergy ucertaity of a Gaussia wave packet is combied with the agular frequecy relatio Eergy-Time Ucertaity Priciple:. The Ucertaity Relatios i 3 Dimesios 5.7: Probability, Wave Fuctios, ad the Copehage Iterpretatio The wave fuctio determies the likelihood (or probability) of fidig a particle at a particular positio i space at a give time (i.e. time-idepedet). The total probability of fidig the electro is 1. Forcig this coditio o the wave fuctio is called ormalizatio.
7 The Copehage Iterpretatio 5.8: Particle i a Box See i Ch.6 Bohr s iterpretatio of the wave fuctio cosisted of 3 priciples: 1) The ucertaity priciple of Heiseberg 2) The complemetarity priciple of Bohr 3) The statistical iterpretatio of Bor, based o probabilities determied by the wave fuctio A particle of mass m is trapped i a oe-dimesioal box of width l. The particle is treated as a wave. The box puts boudary coditios o the wave. The wave fuctio must be zero at the walls of the box ad o the outside. I order for the probability to vaish at the walls, we must have a itegral umber of half wavelegths i the box. Together these three cocepts form a logical iterpretatio of the physical meaig of quatum theory. Accordig to the Copehage iterpretatio, physics depeds o the outcomes of measuremet. The eergy of the particle is. The possible wavelegths are quatized which yields the eergy: The possible eergies of the particle are quatized. Probability of the Particle See i Ch.6 The probability of observig the particle betwee x ad x + dx i each state is Note that E 0 = 0 is ot a possible eergy level. The cocept of eergy levels, as first discussed i the Bohr model, has surfaced i a atural way by usig waves.
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