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Deriving the Feynman Path Integral For the One-Dimensional Free Particle and Simple Harmonic Oscillator
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| Organization: | Physicist at Large Consulting |
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2007-06-08
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The Feynman Path Integral is a way of calculating the quantum-mechanical propagator G( xb, tb, xa, ta ), which gives the probability amplitude for a particle at position xb and time tb, in terms of the probability amplitude at position xa and time ta.
The standard method for calculating the propagator is solution of the appropriate partial differential equation, for example, the Schrodinger equation for non-relativistic quantum mechanics. The Feynman Path Integral approach calculates the propagator by summing exp( i S/ hbar ) over all possible paths from xa ta to xb tb , where S is the action, i is the square root of -1 and hbar is Planck's constant divided by 2 Pi. For some simple situations, the Path Integral can be calculated from the action of the classical physics path, by assuming that the Path Integral is equal to exp( i Sclassical /hbar ) multiplied by a function of t' - t, where Sclassical is the action of the classical path. This is the approach taken here.
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Feynman, Path Integral, Propagator, Free Particle, Simple Harmonic Oscillator
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| Feynman Path Integral-1.nb (51 KB) - Mathematica Notebook |
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