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Influence of the wave packet shape to its time development

Géza I. Márk

November 20, 1997

Research Institute for Materials Science, H-1525 Budapest, P.O.Box 49, Hungary
Tel.: +36-1-3959220
Fax: +36-1-3959284

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An analysis of the behaviour of different wave packets is given in one dimensional free space. Several typical wave packet shapes are investigated: Gaussian, Dirac delta, plane wave and different compact supported ones. Cases of jumps in the wave function, jumps in its derivative and the infinitely smooth case (all derivatives are continuous) are analysed. The spectral properties and the decay form of the momentum space wave function depend on the continuity properties of the wave packet in coordinate space. The initial shape of the wave packet determines its time development. In non-relativistic quantum mechanics all wave packets with bounded support spread to the whole space infinitely fast. It is shown that while the Gaussian minimises the Heisenberg uncertainty relation, for our other wave packets $\Delta x \cdot \Delta k \gt \hbar / 2$. The time dependence of $\Delta x$ is described by a simple formula universal in the sense that the value of the spreading speed depends on the specific wave function shape only through the value of $\Delta k$.

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Last updated: Dec 4, 97 Géza I. Márk