3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fund of Mat Sci: Bonding – Lecture 5/6THE HYDROGEN ATOMComic strip removed for copyright reasons.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Last Time• Metal surfaces and STM• Dirac notation• Operators, commutators, some postulates3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Homework for Mon Oct 3• Study: 18.4, 18.5, 20.1 to 20.5.• Read – before 3.014 starts next week:22.6 (XPS and Auger)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Second Postulate• For every physical observable there is a corresponding Hermitian operator3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Hermitian Operators1. The eigenvalues of a Hermitian operator are real 2. Two eigenfunctions corresponding to different eigenvaluesare orthogonal3. The set of eigenfunctions of a Hermitian operator is complete4. Commuting Hermitian operators have a set of common eigenfunctions3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)The set of eigenfunctions of a Hermitianoperator is completeFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Third Postulate• In any single measurement of a physical quantity that corresponds to the operator A, the only values that will be measured are the eigenvalues of that operator.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Position and probabilityGraphs of the probability density for positions of a particle in a one-dimensional hard box according to classical mechanics removed for copyright reasons.Graph of the probability density for positions of a particle in a one-dimensional hard box removed for copyright reasons.See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 554, figure 15.2.See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 555, figure 15.3.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Quantum double-slitSource: Wikipedia3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Quantum double-slitAbove: Thomas Young's sketch of two-slit diffraction of light. Narrow slits at A and B act as sources, and waves interfering in various phases are shown at C, D, E, and F. Source: WikipediaImage 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)Fourth Postulate• If a series of measurements is made of the dynamical variable A on an ensemble described by Ψ, the average (“expectation”) value is ΨΨΨΨ=AAˆ3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)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.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Spherical Coordinatessin cossin sincosxryrzrθϕθϕθ===zθ0φPyr = rxFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3-d IntegrationDiagram of an infinitesimal volume element in spherical polarcoordinates removed for copyright reasons.See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 1006, figure B.4.Angular MomentumClassical QuantumLrp=×rrr3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Commutation Relation2222222ˆˆˆˆˆˆ ˆˆ ˆˆ,,,0ˆˆ ˆ,0xyzxyzxy zLLLLLL LL LLLL iL=++⎡⎤⎡⎤⎡⎤===⎣⎦⎣⎦⎣⎦⎡⎤=≠⎣⎦hAngular Momentum in Spherical Coordinates22222ˆ11ˆsinsin sinzLiLϕθθθθθϕ∂=−∂⎛⎞∂∂∂⎛⎞=− +⎜⎟⎜⎟∂∂ ∂⎝⎠⎝⎠hh3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Simultaneous eigenfunctions of L2, Lz()()() ()()22ˆ,,ˆ,1,mmzl lmmllLY m YLY l l Yθϕ θϕθϕθϕ==+hh()()(),mmllmYθϕθϕ=Θ Φ3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)Spherical Harmonics in Real FormFigure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)An electron in a central potential (I)2222222222 2222 222ˆ( ) needs to be in spherical coordinates211 1ˆsin ( )2sinˆ1ˆ()2sineeeLHrHVrmHrVrmr rVrmr r rrrrrϑϑϑ ϑ ϑϕ=− ∇ + ∇⎡⎤⎡⎤∂∂∂∂ ∂ ∂ ∂⎛⎞ ⎛ ⎞=− + + +⎢⎜⎟ ⎜ ⎟ ⎥∂∂ ∂ ∂ ∂⎝⎠ ⎝ ⎠⎣⎦⎛⎞=− − +⎢⎥⎜⎟∂∂⎝⎠⎣⎦hhhhr*3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)An electron in a central potential (II)22222ˆ1ˆ()22eedd LHrVrm r dr dr m r⎛⎞=− + +⎜⎟⎝⎠h() ()(, )rRrYψϑϕ=r3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)An electron in a central potential (III)222221(1)() () ()22nl nl nleedd llrVrRrERrm r dr dr m r⎡⎤+⎛⎞−++=⎢⎥⎜⎟⎝⎠⎣⎦hhWhat is the V(r) potential ?3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)21-1-21 2 3 4 5 6Vcentripetal(r)1010r(m)Veff(r)VCoulomb(r)1018v(r) (J)Figure by MIT OCW.The Radial Wavefunctionsfor Coulomb V(r)3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)221100 3 4R10rrrr-0.20000.20.40.040.080.120.62rrr6 102 6 100000R20R21R30R31R324488121216164 8 12 16-0.10.20.4-0.040.040.080.020.04Radial functions Rnl(r) and radial distribution functions r2R2nl(r) foratomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.Figure by MIT OCW.3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)The Radial Density221100 3 4R10000.20.41 2
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