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Journal of Qufu Normal University

DISCUSSIONS ON POTENTIAL-WELL PROBLEM

Wei Yan

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The bounded states and the space possibility density distribution for the one_dimension potential_well problem are discussed, some relations to the possibility conservation and the parallelism principle are given.

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We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.
The paper proposes a new method of describing physical reality using the mathematical apparatus of integer theory. Applying this apparatus becomes possible after introducing the principle of commensurability of conserved quantities into physics. It states that at the interaction of physical objects, only such exchanges of conserved quantities (energies, momentums, etc.) are possible so that these quantities have a common measure before and after the interaction of objects. Methods of integer theory solve two classical problems of physics: the radiation spectrum of a hydrogen-like atom and the scattering problem. The known results of such solutions are obtained, and new data are also obtained. It follows from the obtained that the physical reason for the ‘allowed’ states of atomic systems is the necessity of commensurability of the excited states of the atom with its ground state in all conserved quantities. This gives physical foundations to the quantum hypothesis of N. Bohr, which it has not had so far. The strict causal relation between stationary quantum states of atomic systems is also established, i.e. the principle of causality in the quantum world is restored.