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PathIntegrals

# PathIntegrals - Path Integrals in Quantum Mechanics Michael...

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Path Integrals in Quantum Mechanics Michael Fowler 10/24/07 Huygen’s Picture of Wave Propagation If a point source of light is switched on, the wavefront is an expanding sphere centered at the source. Huygens suggested that this could be understood if at any instant in time each point on the wavefront was regarded as a source of secondary wavelets, and the new wavefront a moment later was to be regarded as built up from the sum of these wavelets. For a light shining continuously, this process just keeps repeating. New wave front slightly later Huygens’ picture of how a spherical wave propagates: each point on the wave front is a source of secondary wavelets that generate the new wave front. Wave front at time t Sample secondary wavelets What use is this idea? For one thing, it explains refraction—the change in direction of a wavefront on entering a different medium, such as a ray of light going from air into glass. If the light moves more slowly in the glass, velocity v instead of c , with v < c , then Huygen’s picture explains Snell’s Law, that the ratio of the sines of the angles to the normal of incident and transmitted beams is constant, and in fact is the ratio c / v . This is evident from the diagram below: in the time the wavelet centered at A has propagated to C , that from B has reached D , the ratio of lengths AC / BD being c / v . But the angles in Snell’s Law are in fact the angles ABC , BCD , and those right-angled triangles have a common hypotenuse BC , from which the Law follows.

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2 D C W B air glass W A B A Huygens’ explanation of refraction: showing two wavelets from the wavefront AB. W B is slowed down compared with W A , since it is propagating in glass. This turns the wave front through an angle. Fermat’s Principle of Least Time We will now temporarily forget about the wave nature of light, and consider a narrow ray or beam of light shining from point A to point B , where we suppose A to be in air, B in glass. Fermat showed that the path of such a beam is given by the Principle of Least Time: a ray of light going from A to B by any other path would take longer. How can we see that? It’s obvious that any deviation from a straight line path in air or in the glass is going to add to the time taken, but what about moving slightly the point at which the beam enters the glass? 2 θ glass B 1 air A Where the air meets the glass, the two rays, separated by a small distance CD = d along that interface, will look parallel:
3 AC = d 1 sin θ C B BD = 2 sin d 1 2 ray 2 A D Magnified view of 2 rays passing through interface: ray 1 is the minimum time path. Rays encounter interface distance CB = d apart. ray 1 (Feynman gives a nice illustration: a lifeguard on a beach spots a swimmer in trouble some distance away, in a diagonal direction. He can run three times faster than he can swim. What is the quickest path to the swimmer?) Moving the point of entry up a small distance d , the light has to travel an extra 1 sin d in air, but a distance less by 2 sin d in the glass, giving an extra travel time 12 sin / sin / td

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PathIntegrals - Path Integrals in Quantum Mechanics Michael...

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