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Separability and invariance properties in non-relativistic and relativistic quantum mechanics

Skiff N. Sokolov

Published Jun 1, 1975• 0 citations
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Separability and invariance in nonrelativstic and relativistic quantum mechanics

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Energy analysis of four dimensional extended hyperbolic Scarf I plus three dimensional separable trigonometric noncentral potentials using SUSY QM approach

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The non-relativistic energies and wave functions of extended hyperbolic Scarf I plus separable non-central shape invariant potential in four dimensions are investigated using Supersymmetric Quantum Mechanics (SUSY QM) Approach. The three dimensional separable non-central shape invariant angular potential consists of trigonometric Scarf II, Manning Rosen and Poschl-Teller potentials. The four dimensional Schrodinger equation with separable shape invariant non-central potential is reduced into four one dimensional Schrodinger equations through variable separation method. By using SUSY QM, the non-relativistic energies and radial wave functions are obtained from radial Schrodinger equation, the orbital quantum numbers and angular wave functions are obtained from angular Schrodinger equations. The extended potential means there is perturbation terms in potential and cause the decrease in energy spectra of Scarf I potential.

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We illustrate Craig Dilworth’s conception of scientific explanation in discussing the interpretation of relativistic quantum mechanics (rather than non–relativistic quantum mechanics, which misses significant elements). The Lorentz–invariant macro–relativity of 1905, which preserved temporal succession and was compatible with the concept of a retarded causality, is distinguished from the Lorentz–and-Lüders invariant micro–relativity of 1952, which postulates a reversibility (CPT–invariance) of causality. In 1927 Einstein brought into prominence an essential incompatibility between his relativity of 1905 (supplemented by the postulate of retarded causality) and ‘the statistical interpretation’ of the ‘new quantum mechanics’ of Born (1926) and Jordan (1926). Here we show (in keeping with other authors) the perfect compatibility of micro–relativity (in the sense of Lüders) and the ‘wavelike calculus of probabilities’ according to Born and Jordan; the ‘EPR correlations’ and ‘Wheeler’s smoky dragon’ illustrate the spatial and temporal aspects of the ‘non–separability’ inherent in quantum mechanics. We argue that psychic non–separability is a corollary of physical non–separability. It thus appears that telepathy, precognition and psychokinesis are more than permitted – they are actually suggested by the symmetries of the formalism. Incidentally, we propose a reconceptualization and a reformalization of the classical calculus of conditional probabilities intended to confer on them an intersubjectivity in harmony with a contingent physics, a manifest relativistic covariance, and an exact correspondence with the ‘wave’ formalism of Born and Jordan.

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Relativistic invariance in quantum mechanics

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Il Nuovo Cimento•Mar 1, 1956•44 citations
The non-relativistic energies and wave functions of extended hyperbolic Scarf I plus separable non-central shape invariant potential in four dimensions are investigated using Supersymmetric Quantum Mechanics (SUSY QM) Approach. The three dimensional separable non-central shape invariant angular potential consists of trigonometric Scarf II, Manning Rosen and Poschl-Teller potentials. The four dimensional Schrodinger equation with separable shape invariant non-central potential is reduced into four one dimensional Schrodinger equations through variable separation method. By using SUSY QM, the non-relativistic energies and radial wave functions are obtained from radial Schrodinger equation, the orbital quantum numbers and angular wave functions are obtained from angular Schrodinger equations. The extended potential means there is perturbation terms in potential and cause the decrease in energy spectra of Scarf I potential.
We illustrate Craig Dilworth’s conception of scientific explanation in discussing the interpretation of relativistic quantum mechanics (rather than non–relativistic quantum mechanics, which misses significant elements). The Lorentz–invariant macro–relativity of 1905, which preserved temporal succession and was compatible with the concept of a retarded causality, is distinguished from the Lorentz–and-Lüders invariant micro–relativity of 1952, which postulates a reversibility (CPT–invariance) of causality. In 1927 Einstein brought into prominence an essential incompatibility between his relativity of 1905 (supplemented by the postulate of retarded causality) and ‘the statistical interpretation’ of the ‘new quantum mechanics’ of Born (1926) and Jordan (1926). Here we show (in keeping with other authors) the perfect compatibility of micro–relativity (in the sense of Lüders) and the ‘wavelike calculus of probabilities’ according to Born and Jordan; the ‘EPR correlations’ and ‘Wheeler’s smoky dragon’ illustrate the spatial and temporal aspects of the ‘non–separability’ inherent in quantum mechanics. We argue that psychic non–separability is a corollary of physical non–separability. It thus appears that telepathy, precognition and psychokinesis are more than permitted – they are actually suggested by the symmetries of the formalism. Incidentally, we propose a reconceptualization and a reformalization of the classical calculus of conditional probabilities intended to confer on them an intersubjectivity in harmony with a contingent physics, a manifest relativistic covariance, and an exact correspondence with the ‘wave’ formalism of Born and Jordan.