MIT 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 Fall 2007 – Lecture 4 CLOSE TO COLLAPSE •The collapse of the wavefunction 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Travel • Office hour (this time only):Office hour (this time only): – This Friday, Sep 21, 4pm– (instread of Mon, Sep 24, 4pm) 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 1Last time: Wave mechanics 11. The ket Ψ describe the systemTheket Ψ describe the system 2. The evolution is deterministic, but it applies to stochastic events 3. Classical quantities are replaced by operators 4. The results of measurements are eigenvalues, dth ktll i i tand the ket collapses in an eigenvector 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Commuting Hermitian operators have a set of common eigenfunctions 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 2•Fifth postulate • If the measurement of the physicalIf the measurement of the physical quantity A gives the result an , the wavefunction of the system immediately after the measurement is the eigenvector nϕ 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Position and probability 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 3 Diagram 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 aparticle in a one-dimensional hard box according to classical mechanics removed for copyright reasons. See Mortimer, R. G. Physical Chemistry. 2nd San Diego, CA: Elsevier, 2000, page 555, Figure 15.3.ed.“Collapse” of the wavefunction 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Quantum double-slit 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.gif 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 4 Image of a double-slit experiment simulation removed due to copyright restrictions. Please see "Double Slit Experiment." in Visual Quantum Mechanics.x(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.•Deterministic vs. stochastic • Classical macroscopic objects: we have well-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 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. 5Cat wavefunction 1 1 ⎜⎝exp⎛ − exp ⎛ −⎜⎝ ⎛ ⎞ ⎛ ⎞ ⎞⎟⎠ ⎞⎟⎠ 2 2t t Ψ ()t = Ψalivecat + Ψdead 1− ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ τ τ • There is not a value of the observable until it’s measured (a conceptually different “statistics” from thermodynamics) 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Uncertainties, and Heisenberg’s Indetermination Principle ( )2 ( )22 2∆A = A − A = A − A 1∆A∆B ≥ [A, B]22 di⎡ ⎢⎣ x ,− ⎤ i h h ⎥⎦ = dx 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 6Albert Einstein: “Gott wurfelt nicht!” [God doesAlbert Einstein: Gott wurfelt [God doesErwin Schrödinger: “Had I known that we were Erwin Had I known we wereLinewidth Broadening Image 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. ∆E∆t ≥ h 2 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Top Three List • • nicht! not play dice!] • Werner Heisenberg “I myself . . . only came to believe in the uncertainty relations after many pangs of conscience. . .” • • Schrödinger: that 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) 7Spherical Coordinates = sinθcos xr ϕ = sinθsin yr ϕ zr= cos θθ 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Angular Momentum ClassicalClassical QuantumQuantum r rr ⎛⎞LrpLyxzˆˆpzˆˆ y = −h⎜⎟−z=× ˆ =−p iy ∂∂ zy∂∂⎝⎠ ⎛⎞⎛⎞Lˆ =−zp xˆˆˆˆ p i z = − h ∂∂∂∂− xyxz ⎜⎟xz∂∂⎝⎠ ∂∂Lxˆˆpyˆˆ p ix − h −yˆ =−=⎛⎞zyx ⎜⎟yx∂∂⎝⎠ 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) 8 zθ0φPyr = rxFigure by MIT OpenCourseWare.⎣ ⎦=⎜ ⎟h⎛ ⎞Commutation Relation ˆ22 = ˆ22 + ˆ22 + ˆ22LLLL x y z ˆˆ2 ˆˆ2 ˆˆ⎡LL ⎡LL ⎡ 2, z⎣ , x ⎤⎦ = ⎣ , y ⎤⎦ = ⎣LL⎤⎦ =0 ⎡ ˆˆ ⎤⎤ = ˆ⎡LLx, y h ⎣ LL ⎦ iL ihL≠≠00 z 3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Angular Momentum in Spherical Coordinates Lˆ =−ih ∂ z∂ϕ Lˆ22 =−h 22 ⎛ 11 ∂∂ ⎛⎛ siinθθ ∂∂ ⎞⎞ + 11 ∂∂2 ⎞ ⎜⎝