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- Thread starter Theage
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In summary, the free particle will move from x to y in time t as U(x,y,t)=\frac{d^3p}{(2\pi)^3}\int\frac{d^3p}{(2\pi)^3}e^{i(p^2/2m)t}e^{i\vec{p}\cdot\Delta x}=\int\frac{d^3p}{(2\pi)^3}e^{i(p^2/2m)t}e^{ip\Delta x\cos\theta}. After performing the angular integrations (which are both trivial), U(x,y,t) will be

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Thanks! Amazingly the [itex]2\pi[/itex]'s work out using Gaussian integral methods.

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And there should be a minus sign in the exponential with the time!

A nonrelativistic free particle propagator is a mathematical function that describes the probability amplitude of a particle to travel from one point in space to another in a nonrelativistic system. It takes into account the initial and final positions of the particle, as well as the time it takes for the particle to travel between the two points.

A nonrelativistic free particle propagator is calculated using the Feynman path integral, which sums over all possible paths that the particle can take between the initial and final positions. The propagator is then the integral of this sum over all possible times.

The nonrelativistic free particle propagator is an important tool in quantum mechanics as it allows us to calculate the probability of a particle to travel between two points in a nonrelativistic system. It is also used in calculating transition amplitudes and cross sections in scattering experiments.

The main difference between the nonrelativistic and relativistic free particle propagators lies in the treatment of time. The nonrelativistic propagator assumes a fixed, universal time for all observers, while the relativistic propagator takes into account the time dilation and length contraction effects of special relativity.

The nonrelativistic free particle propagator has various applications in quantum mechanics, such as in calculating the probability of a particle to tunnel through a potential barrier, or in predicting the behavior of particles in a double-slit experiment. It also has practical applications in fields such as material science and quantum computing.

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