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Wave packet with special relativity demonstrating quantum rules, Schrödinger's equation and propagator integral

Wave packet with special relativity demonstrating quantum rules, Schrödinger's equation and propagator integral

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The physics of special relativity is applied to a wave-packet model of a particle without the usual de Broglie-like assumption of Planck's rule, E = hv. The velocity of the particle is equated to the group-velocity of the wave-packet to associate the particle with its wave-packet. With such an identification it is demonstrated using special relativity that the momentum four-vector, P = [p, E/c2] of the particle is parallel and proportional to a central wave-number four-vector, K0 = [k0. ω0/c2], of the particle's wave-packet. The constant of proportionality, relating these two four-vectors, has the dimension of action and by diffraction experiments it has the value h/2π where h is Planck's constant. This refinement of the de Broglie wave-like nature of a particle suggests that special relativity underlies quantum physics more deeply than heretofore believed. The above quantum rules are used to find the wave-packet function in the one-dimensional case of a non-relativistic free particle and a particle in a conservative force field of potential V (x). This wave-packet function is shown to satisfy an integral time-evolution equation which relates the wave function at time t0 to the function at a slightly later time t. The resulting integral equation is equivalent to the propagator equation or path-integral hypothesised by Feynman. This fact also demonstrates, using the method of Feynman, that the wave-packet function satisfies Schrödinger's partial differential equation. The wave-packet function of a particle in a force field is directly shown to satisfy a linear integral equation which relates the wave-packet at one instant of time to its values at all later times. The kernel or propagator of this integral equation is also equivalent to Feynman's path integral. A computation demonstrates that the position of a moving particle in a simple quadratic potential well does not become more uncertain with the passage of time. This contrasts with the behaviour of a moving free particle which as a function of time becomes more and more difficult to accurately locate.

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