Physics – Quantum Physics
Scientific paper
1997-10-02
Physics
Quantum Physics
Latex, 11 pages, extended, new references added
Scientific paper
We modify the Schr\"{o}dinger equation in a way that preserves its main properties but makes use of higher order derivative terms. Although the modification represents an analogy to the Doebner-Goldin modification, it can differ from it quite distinctively. A particular model of this modification including derivatives up to the fourth order is examined in greater detail. We observe that a special variant of this model partially retains the linear superposition principle for the wave packets of standard quantum mechanics remain solutions to it. It is a peculiarity of this variant that a periodic structure emerges naturally from its equations. As a result, a free particle, in addition to a plane wave solution, can possess band solutions. It is argued that this can give rise to well-focused particle trajectories. Owing to this peculiarity, when interpreted outside quantum theory, the equations of this modification could also be used to model pattern formation phenomena.
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