Wave-Particle Duality & Quantum Mechanics: X-Ray Scattering & Uncertainty, Slides of Physics

Download Wave-Particle Duality & Quantum Mechanics: X-Ray Scattering & Uncertainty and more Slides Physics in PDF only on Docsity!  X-Ray Scattering  De Broglie Waves  Electron Scattering  Wave Motion  Waves or Particles?  Uncertainty Principle  Probability, Wave Functions, and the Copenhagen Interpretation  Particle in a Box Wave Properties of Matter and Quantum Mechanics I I thus arrived at the overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. - Louis de Broglie, 1929 docsity.com 5.1: X-Ray Scattering  Max von Laue suggested that if x rays were a form of electromagnetic radiation, interference effects should be observed.  Crystals act as three-dimensional gratings, scattering the waves and producing observable interference effects. docsity.com 5.2: De Broglie Waves  Prince Louis V. de Broglie suggested that mass particles should have wave properties similar to electromagnetic radiation.  The energy can be written as  Thus the wavelength of a matter wave is called the de Broglie wavelength: docsity.com Bohr’s Quantization Condition  One of Bohr’s assumptions concerning his hydrogen atom model was that the angular momentum of the electron-nucleus system in a stationary state is an integral multiple of h/2π.  The electron is a standing wave in an orbit around the proton. This standing wave will have nodes and be an integral number of wavelengths.  The angular momentum becomes: docsity.com 5.3: Electron Scattering  Davisson and Germer experimentally observed that electrons were diffracted much like x rays in nickel crystals.  George P. Thomson (1892–1975), son of J. J. Thomson, reported seeing the effects of electron diffraction in transmission experiments. The first target was celluloid, and soon after that gold, aluminum, and platinum were used. The randomly oriented polycrystalline sample of SnO2 produces rings as shown in the figure at right. docsity.com Principle of Superposition  When two or more waves traverse the same region, they act independently of each other.  Combining two waves yields:  The combined wave oscillates within an envelope that denotes the maximum displacement of the combined waves.  When combining many waves with different amplitudes and frequencies, a pulse, or wave packet, is formed which moves at a group velocity: ugr = Δω / Δk. docsity.com Fourier Series  The sum of many waves that form a wave packet is called a Fourier series:  Summing an infinite number of waves yields the Fourier integral: docsity.com Wave Packet Envelope  The superposition of two waves yields a wave number and angular frequency of the wave packet envelope.  The range of wave numbers and angular frequencies that produce the wave packet have the following relations:  A Gaussian wave packet has similar relations:  The localization of the wave packet over a small region to describe a particle requires a large range of wave numbers. Conversely, a small range of wave numbers cannot produce a wave packet localized within a small distance. docsity.com 5.5: Waves or Particles?  Young’s double-slit diffraction experiment demonstrates the wave property of light.  However, dimming the light results in single flashes on the screen representative of particles. docsity.com Electron Double-Slit Experiment  C. Jönsson of Tübingen, Germany, succeeded in 1961 in showing double-slit interference effects for electrons by constructing very narrow slits and using relatively large distances between the slits and the observation screen.  This experiment demonstrated that precisely the same behavior occurs for both light (waves) and electrons (particles). docsity.com Which slit?  To determine which slit the electron went through: We set up a light shining on the double slit and use a powerful microscope to look at the region. After the electron passes through one of the slits, light bounces off the electron; we observe the reflected light, so we know which slit the electron came through.  Use a subscript “ph” to denote variables for light (photon). Therefore the momentum of the photon is  The momentum of the electrons will be on the order of .  The difficulty is that the momentum of the photons used to determine which slit the electron went through is sufficiently great to strongly modify the momentum of the electron itself, thus changing the direction of the electron! The attempt to identify which slit the electron is passing through will in itself change the interference pattern. docsity.com Energy Uncertainty  If we are uncertain as to the exact position of a particle, for example an electron somewhere inside an atom, the particle can’t have zero kinetic energy.  The energy uncertainty of a Gaussian wave packet is combined with the angular frequency relation  Energy-Time Uncertainty Principle: . docsity.com 5.7: Probability, Wave Functions, and the Copenhagen Interpretation  The wave function determines the likelihood (or probability) of finding a particle at a particular position in space at a given time.  The total probability of finding the electron is 1. Forcing this condition on the wave function is called normalization. docsity.com The Copenhagen Interpretation  Bohr’s interpretation of the wave function consisted of 3 principles: 1) The uncertainty principle of Heisenberg 2) The complementarity principle of Bohr 3) The statistical interpretation of Born, based on probabilities determined by the wave function  Together these three concepts form a logical interpretation of the physical meaning of quantum theory. According to the Copenhagen interpretation, physics depends on the outcomes of measurement. docsity.com