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Ben-Gurion University of the Negev Atomic Atomic and and Molecular Molecular Physics Physics for for Physicists Physicists Ron Folman Chapter 5: Wave mechanics in quantum mechanics: The wave function, the uncertainty principle, statistical nature Main References: Corney A. Atomic and Laser Spectroscopy, Oxford UP, 1987; QC 688.C67 1987; Chapters in Modern Physics, Open University Exercises: Dudi Moravchik. www.bgu.ac.il/atomchip www.bgu.ac.il/nanofabrication www.bgu.ac.il/nanocenter Wave mechanics: The mathematical formalism I. (the wave packet, the uncertainty principle, statistical nature) Vg=dw/dk If particles are waves, how do we use infinite waves to described localized entities? The wave packet has a finite width and can therefore be used to describe particles. How do we make wave packets from waves? We used many frequencies! Ψ(x)=∫dk Ψ(k) eikx Fourier Transform X – group velocity ↑- phase velocity The uncertainty principle ∆x ∆p = h A “naive” way of getting it: ∆x ⋍ nλ nh/p ∆λ λ/n ∆λ = ∆p h/p2 = h/p ∆p/p = λ ∆p/p => ∆λ/λ = ∆p/p 1/n ∆p p/n • a bullet (50g) and an electron are both shot with a speed of 300m/s. This velocity Is measured with an accuracy of 0.01%. With what accuracy can we measure their Position? Graphical example of why this happens: The Bohr/Heisenberg microscope The apparatus The electron must have the transverse momentum uncertainty as the photon (because of momentum conservation). As the photons that we see come from an angle of θ the photon transverse momentum uncertainty is: ∆px = p sinθ = sinθ h/λ Every microscope has the limit (the so-called diffraction limit) of observing a point like particle with a width of ∆x = λ / sinθ . This is then the accuracy With which we know the particles position Exercise: To fight the idea of W2+P2=1, Einstein suggested to place an infinitely sensitive “which path” detector in one of the two slits of the double-slit experiment Based on the uncertainty principle, and the fact that the distance between the two slits should be of order λ (why?), prove that Einstein’s idea cannot work. Exercise: explain diffraction with the FT, and show that you know how to estimate the width of the zero order from the uncertainty principle. Another example of the uncertainty principle: ∆E ∆t > h Example I: the line width of a pulse laser. What is the minimal line width of a Femto second laser? Example II: the line width of an atomic energy level. Why is the line width of the Rubidium excited level 5MHz? • Prove another manifestation of the uncertainty principle: namely, that ∆N ∆Φ > h, where N is the number of particles (FOCK state) and Φ is the phase. The statistical nature of QM: the interpretation of the wave function. The statistical nature of the wave function. Prob.=|ψ|2 where |ψ| is called the amplitude Think of what a two particle wave function means? Ψ(x1,y1,x2,y2,t)