Abstract
The initial-value problem of the force-free Schrodinger equation is solved in closed-form for initial wave functions which are constrained to a finite region of space and improper boundary conditions at infinity. It is shown that the wave function and probability density propagate with infinite speed into the space. This is explained mathematically to be due to the parabolic nature of the Schrodinger equation. It is concluded that the Schrodinger equation gives a qualitatively not quite satisfactory description of the (time-dependent) propagation of probability densities due to the occurrence of infinite propagation speeds.
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