3.23 Fall 2007 – Lecture 4 CLOSE TO COLLAPSETravelLast time: Wave mechanicsCommuting Hermitian operators have a set of common eigenfunctionsFifth postulatePosition and probability“Collapse” of the wavefunctionQuantum double-slitDeterministic vs. stochasticWhen scientists turn bad…Cat wavefunctionUncertainties, and Heisenberg’s Indetermination PrincipleLinewidth BroadeningTop Three ListSpherical CoordinatesAngular MomentumCommutation RelationAngular Momentum in Spherical CoordinatesEigenfunctions of Lz , L2Simultaneous eigenfunctions of L2, LzSpherical Harmonics in Real FormSame as a beating drum……for the career helioseismologistAngular Momentum, then…An electron in a central potential (I)An electron in a central potential (II)An electron in a central potential (III)What is the Veff(r) potential ?The Radial Wavefunctions for Coulomb V(r)The Grand TableSolutions in the central Coulomb Potential: the Alphabet SoupMIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)3.23 Fall 2007 –Lecture 4CLOSE TO COLLAPSEThe collapse of the wavefunctionTravel• Office hour (this time only) 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)Last time: Wave mechanics1. The ket describe the system2. The evolution is deterministic, but it applies to stochastic events3. Classical quantities are replaced by operators4. The results of measurements are eigenvalues, and the ket collapses in an eigenvectorΨ3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)Commuting Hermitian operators have a set of common eigenfunctionsFifth postulate• If the measurement of the physical quantity A gives the result an, the wavefunction of the system immediately after the measurement is the eigenvector 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)nϕ3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Position and probabilityDiagram showing the probability densities of the first 3 energy states in a 1D quantum well of width L.Graphs of the probability density for positions of a particle in a one-dimensionalhard box according to classical mechanics removed for copyright reasons. See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, page 555, Figure 15.3.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)“Collapse” of the wavefunctionx(c)x(a)ax(b)aaThe wave function of a particle in a box. (a) Before a position measurement (schematic). The probabilitydensity is nonzero over the entire box (except for the endpoints). (b) Immediately after the positionmeasurement (schematic). In a very short time, the particle cannot have moved far from the position givenby the measurement, and the probability density must be a sharply peaked function. (c) Shortly after aposition measurement (schematic). After a short time, the probability density can be nonzero over a largerregion.Figure by MIT OpenCourseWare.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Quantum double-slitImage of a double-slit experiment simulation removed due to copyright restrictions. Please see "Double Slit Experiment.“in Visual Quantum Mechanics..Image removed due to copyright restrictions. Please see any experimental verification of the double-slit experiment, such as http://commons.wikimedia.org/wiki/Image:Doubleslitexperiment_results_Tanamura_1.gif3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Deterministic vs. stochastic• Classical, macroscopic objects: we have well-defined values for all dynamical variables at every instant (position, momentum, kinetic energy…)• Quantum objects: we have well-defined probabilities of measuring a certain value for a dynamical variable, when a large number of identical, independent, identically prepared physical systems are subject to a measurement.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)When scientists turn bad…Image from Wikimedia Commons, http://commons.wikimedia.org.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Cat wavefunction• There is not a value of the observable until it’s measured (a conceptually different “statistics” from thermodynamics)()2121exp1exp⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−−Ψ+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−Ψ=Ψττtttdeadalivecat3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Uncertainties, and Heisenberg’s Indetermination Principle()()2222AAAAA −=−=Δ[]BABA ,21≥ΔΔ== idxdix =⎥⎦⎤⎢⎣⎡−,3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Linewidth Broadening2=≥ΔΔ tEImage removed due to copyright restrictions. Please see Fig. 2 in Uhlenberg, G., et al. "Magneto-optical Trapping of Silver Ions." Physical Review A 62 (November 2000): 063404.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Top Three List• Albert Einstein: “Gott wurfelt nicht!” [God does not play dice!]• Werner Heisenberg “I myself . . . only came to believe in the uncertainty relations after many pangs of conscience. . .”• Erwin Schrödinger: “Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business!”3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Spherical Coordinatessin cossin sincosxryrzrθϕθϕθ===zθ0φPyr = rxFigure by MIT OpenCourseWare.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Angular MomentumLrp=×GGGˆˆˆ ˆˆˆˆˆˆˆˆˆˆ ˆˆxzyyxzzyxLypzp iy zzyLzpxp iz xxzLxpyp ix yyx⎛⎞∂∂=−=− −⎜⎟∂∂⎝⎠∂∂⎛⎞=−=− −⎜⎟∂∂⎝⎠⎛⎞∂∂=−=− −⎜⎟∂∂⎝⎠===Classical