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In book "Quantum Mechanics and Path Integral", 3-2 Diffraction through the slit:

Under the fig. 3-3, why did Feynman say that we cannot approach the problem by a single application of the free-particle law motion, since the particle is actually constrained by the slit?

And why Feynman use Gaussian slit?

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I don't have a copy of the book, so can't comment on the Gaussian slit, but you can't use the free particle propagation equation in the presence of a slit, because the slit (a conventional slit) can be modelled by a rectangular shaped potential function. Straightforward free particle propagation only applies when the potential function vanishes, so is inapplicable here.

For anyone else who may not have the book, Feynman's original paper has a very readable treatment of the path integral and is available here.

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