3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Bonding – Lecture 3GHOST IN THE MACHINEImage of a quantum mirage produced by a Co atom placed in the focus of a Co elliptical corral, removed for copyright reasons. Don Eigler, IBM Almaden, Nature (2000). See http://domino.watson.ibm.com/comm/pr.nsf/pages/rscd.quantummirage-picb.html/$FILE/mirage2.jpg@Bobby Douglas, from photo.net3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last time: Schrödinger equation 1. Time-dependent Schrödinger equation for one electron in a potential V(r,t) (a plane wave satisfies this eqn.)2. For a stationary potential V(r), we introduced the method of separation of variables, and obtained a) the stationary Schrödinger equation for the spatial part φ(x), and b) the equation for the time-dependent function f(t)3. Homework: for a free particle it is easy to obtain φ(x) and f(t), and one obtains back the equation of a plane wave4. Studied a free particle in an infinite well (particle in a box)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Fri 16• Study: 15.3 (2-,3-dim box), 16.3 (π-electrons in conjugated molecules), 16.5-6 (scanning tunnelling microscope)• Optional read: 1986 Nobel lecture by Binnig and Rohrer (on the MIT server)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Physical Observables from Wavefunctions• Eigenvalue equation: (the operator is obtained via the “correspondence” principle)• Expectation values for the operator (energy)222() () ()2ExVxxdxmxϕϕ∗⎡⎤∂=−+⎢⎥∂⎣⎦∫h222() () () 2dVx x E xmdxϕϕ⎡⎤−+ =⎢⎥⎣⎦h3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Normalization3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Infinite Square Well222()()2dxExmdxϕϕ−=hFigure by MIT OCW.201816141210864200.0 0.2 0.4 0.6 0.8 1.0x/aEnergy or wave function value3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Infinite Square Well20181614121086420Energy Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Absorption Lines (atomic units)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)The power of carrots• β-carotenePhoto courtesy of Andrew Dunn.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Particle in a 2-dim box22 222(, ) (, )2xy E xymx yϕϕ⎛⎞∂∂−+ =⎜⎟∂∂⎝⎠h3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Particle in a 2-dim box(, ) sin sinlx myxy Cabππϕ⎛⎞⎛ ⎞=⎜⎟⎜ ⎟⎝⎠⎝ ⎠22 2228hl mEma b⎛⎞=+⎜⎟⎝⎠3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Particle in a 3-dim box: Farbe defect in halides (e-bound to a negative ion vacancy)Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)From Carl Zeiss to MIT…Scanned image of a journal article removed for copyright reasons. See Avakian, P. and A. Smakula. “Color Centers in Cesium Halide Single Crystals.” Physical Review 120, no. 6 (December 15, 1960).3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Light absorption/emissionMIT Research: Bawendi, Mayes, StellacciNicola Marzari:Porphyrin from http://www.chem.uit.no/KJEMI/publications2.html, Raman spectraFrom Mauri and Lazzeri Phys. Rev. Lett. PaperNicola Marzari:Porphyrin from http://www.chem.uit.no/KJEMI/publications2.html, Raman spectraFrom Mauri and Lazzeri Phys. Rev. Lett. PaperCourtesy of Felice Frankel. Used with permission.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Image removed for copyright reasons.See the chart of various diamondoids at http://www.physik.tu-berlin.de/cluster/diamondoids.html.Scanned image of a journal article removed for copyright reasons. See Willey, T. M. et al. “Molecular Limits to the Quantum Confinement Model in Diamond Clusters.” Physical Review Letters 95 (September 9,
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