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Hojman, S. A., & Asenjo, F. A. (2020). A new approach to solve the one-dimensional Schrodinger equation using a wavefunction potential. Phys. Lett. A, 384(36), 7 pp.
Abstract: A new approach to find exact solutions to one-dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non-vanishing Bohm potentials. For most of the potentials, no solutions to the Schrodinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non-vanishing Bohm potentials. New examples of this kind are found and discussed. (C) 2020 Elsevier B.V. All rights reserved.
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Hojman, S. A., & Asenjo, F. A. (2020). Classical and Quantum Dispersion Relations. Phys. Scr., 95(8), 7 pp.
Abstract: It is showed that, in general, classical and quantum dispersion relations are different due to the presence of the Bohm potential. There are exact particular solutions of the quantum (wave) theory which obey the classical dispersion relation, but they differ in the general case. The dispersion relations may also coincide when additional assumptions are made, such as WKB or eikonal approximations, for instance. This general result also holds for non-quantum wave equations derived from classical counterparts, such as in ray and wave optics, for instance. Explicit examples are given for covariant scalar, vectorial and tensorial fields in flat and curved spacetimes.
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Hojman, S. A., & Asenjo, F. A. (2020). Dual wavefunctions in two-dimensional quantum mechanics. Phys. Lett. A, 384(13), 5 pp.
Abstract: It is shown that the Schrodinger equation for a large family of pairs of two-dimensional quantum potentials possess wavefunctions for which the amplitude and the phase are interchangeable, producing two different solutions which are dual to each other. This is a property of solutions with vanishing Bohm potential. These solutions can be extended to three-dimensional systems. We explicitly calculate dual solutions for physical systems, such as the repulsive harmonic oscillator and the two-dimensional hydrogen atom. These dual wavefunctions are also solutions of an analogue optical system in the eikonal limit. In this case, the potential is related to the refractive index, allowing the study of this two-dimensional dual wavefunction solutions with an optical (analogue) system. (C) 2020 Elsevier B.V. All rights reserved.
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