for a 1D free particle with initial wave function [itex]\phi(x')[/itex] square shaped(e.g. [itex]\phi(x')=1,x'\in [a,b][/itex],otherwise it vanishes),(adsbygoogle = window.adsbygoogle || []).push({});

my question is: how does it evolve with time [itex]t[/itex]?

if we deal with it in [itex]P[/itex] basis, it is easily solved, using the propagator [itex]U(t)=∫|p'><p'|e^{-\frac{ip'^2 t}{2m\hbar}}dp'[/itex];

but if we directly solve SE in [itex]X[/itex] basis, where [itex]P[/itex] must be written as [itex]-i\frac{∂}{∂x'}[/itex], the initial wavefunction is not continous, so the equation becomes improper at the ends of the interval[itex][a,b][/itex],

so why dose the SE equation seems so distinct in these 2 representations? what goes wrong in [itex]X[/itex] representation?

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# Square shape wave packet spreading

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