3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Bonding – Lecture 1 bisWAVE MECHANICSPhoto courtesy of Malene Thyssen, www.mtfoto.dk/malene/3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last Time1. Players: particles (protons and neutrons in the nuclei, electrons) and electromagnetic fields (photons)2. Forces: electromagnetic3. Dynamics: Newton (macroscopic), Maxwell (fields), Schrödinger (microscopic)4. De Broglie relation λ •p = h3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Mon 12• You know by now: 12.5, 13.2, 13.3 • Study: 13.4 and 13.5•Notes on harmonic oscillator -- Section 14.1 in Mortimer, R. G.Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Harmonic Oscillator (I)StationaryobjectSpringMassEquilibriumposition ofmasszz0A mass on a spring. This system can berepresented by a harmonic oscillator.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Harmonic Oscillator (II)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Harmonic Oscillator (III)Graph of the behavior of a harmonic oscillatorremoved for copyright reasons.See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 495, figure 14.2.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)The total energy of the system• Kinetic energy K• Potential energy V3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Polar RepresentationDiagram of the Argand plane removed for copyright reasons.See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 1011, figure B.6.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)A Traveling “Plane” Wave(,) exp[( )]rt A ik r tωΨ= ⋅−rrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Principle of Linear SuperpositionPhoto courtesy of Spiralz3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Wave-particle Duality• Waves have particle-like properties:– Photoelectric effect: quanta (photons) are exchanged discretely– Energy spectrum of an incandescent body looks like a gas of very hot particles Diagrams of the photoelectric effect and of a P-N solar cell, removed for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Wave-particle Duality• Particles have wave-like properties:– Electrons in an atom act like standing waves (harmonics) in an organ pipe– Electrons beams can be diffracted, and we can see the fringes (Davisson and Germer, at Bell Labs in 1926…)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)When is a particle like a wave ?Wavelength • momentum = Planck↕λ •p = h( h = 6.626 x 10-34J s = 2π a.u.)See animation at http://www.kfunigraz.ac.at/imawww/vqm/movies.htmlSelect “Samples from Visual Quantum Mechanics” > “Double-slit experiment”Image of the double-slit experiment removed for copyright reasons. See the simulation at http://www.kfunigraz.ac.at/imawww/vqm/movies.html: "Samples from Visual Quantum Mechanics": "Double-slit Experiment."3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Time-dependent Schrödinger’s equation(Newton’s 2ndlaw for quantum objects)ttritrtrVtrm∂Ψ∂=Ψ+Ψ∇−),(),(),(),(222rhrrrh1925-onwards: E. Schrödinger (wave equation), W. Heisenberg (matrix formulation), P.A.M. Dirac (relativistic)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Plane waves as free particlesOur free particle satisfies the wave equation:)](exp[),( trkiAtrω−⋅=Ψrrrttritrm ∂Ψ∂=Ψ∇−),(),(222rhrh(provided )mkmpE22222hh ===ω3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Stationary Schrödinger’s Equation (I)ttritrtrVtrm∂Ψ∂=Ψ+Ψ∇−),(),(),(),(222rhrrrh*3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Stationary Schrödinger’s Equation
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