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Summary

In this paper we have analysed the effects of the shape of the coordinate space initial wave function on the various properties of the quantum mechanical state. The main effects are the following:

Decay form of the spectral density: depends on the continuity of the derivatives of the wave packet. If the coordinate wave function is infinitely smooth than the momentum wave function has a faster than polynomial decay. In case of jumps in $\psi(x)$ or its derivatives $\varphi(k)$ has a polynomial decay.
Free space time evolution of wave packets with bounded support: leads to an infinite fast spreading speed in non-relativistic QM.
Uncertainty product: is shown to be 1/2 for the Gaussian packet and larger than 1/2 for the other packets.
Asymptotic spreading speed: is constant in free space and its value is equal to the initial momentum dispersion.

We believe that the study of the shape effects of wave packets leads to a deeper understanding of quantum mechanics.

Acknowledgments

This work was partially supported by the Hungarian OTKA Grant No. F 014236.

Next: References Up: Influence of the wave Previous: Perfect localisation in momentum