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Hadronic J.; (United States)

Complex Schroedinger configuration plane and its relation to a classical and real two dimensional representation for one dimensional quantum theory

James G. Gilson

Published Jun 1, 1982• 0 citations
OpenAlex

Abstract

It is indicated how a two dimensional alternative to one dimensional Schroedinger quantum theory is related to nonpotential Hamiltonian theory. The ability of this two dimensional alternative to generate classical electromagnetic and thermal images for the processes taking place on and off the real configuration axis, is explored. In particular, it is shown how the quantum potential is related to a deformation of the rest mass distribution over the two dimensional surface.

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It is indicated how a two dimensional alternative to one dimensional Schroedinger quantum theory is related to nonpotential Hamiltonian theory. The ability of this two dimensional alternative to generate classical electromagnetic and thermal images for the processes taking place on and off the real configuration axis, is explored. In particular, it is shown how the quantum potential is related to a deformation of the rest mass distribution over the two dimensional surface.
This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and momentum observables are taken with respect to one-particle states and their time-dependence is discussed. A potential is added to the free-particle Hamiltonians and q-analogs of the Ehrenfest theorem are derived from the Heisenberg equations of motion. The conservation of probability is proved.
A complete solution to the long standing problem of basing Schroedinger quantum theory on standard stochastic theory is given. The solution covers all "single" particle three-dimensional Schroedinger theory linear or nonlinear and with any external potential present. The system is classical, set in a six-dimensional space and involves vacuum polarization as the background process. Basic vacuum polarization energy characterised oscillators are identified and then in assemblies are analysed in terms of energy occurrence frequencies. The orbits of polarization monopoles are given and shown to be elliptical on subspaces surfaces. The basic process takes place at the speed of light and is of a statistical "zitterbewegung" character. The orthodox quantum probability density bilinear quadratic form is derived from angular momentum consideration within the system which is shown to be a generalisation of the usual quantum structure. Two statistical assemblies are identified, a linear one associated with superposition of eigenfunctions and a quadratic one associated with interactions between eigenstates. It is suggested that this two-tier probabilistic system will remove some possible paradoxes that plague the orthodox thoory. The relation of vacuum polarization in this work with its occurrence in other physical contexts and a connection with "spin" is discussed.
With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H ( x , p ) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, E i , turns out to be zero for all potentials V ( x ) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.
Classical edge detection algorithms often struggle to process large, high-resolution image datasets efficiently. Quantum image processing offers a promising alternative, but current implementations face significant challenges, such as time-consuming data acquisition, complex device requirements, and limited real-time processing capabilities. This work presents a novel paired transform-based quantum representation for efficient image processing. This representation enables the parallelization of convolution operations, simplifies gradient calculations, and facilitates the processing of one-dimensional and two-dimensional signals. We demonstrate that our approach achieves improved processing speed compared to classical methods while maintaining comparable accuracy. The successful implementation of real-world images highlights the potential of this research for large-scale quantum image processing, architecture-specific optimizations, and applications beyond edge detection.
By re-expressing the generalized Schrodinger equation in its coordinate representation in terms of amplitude and phase of the state vector, a quantum-mechanical change of a system can be made to correspond to an ensemble of classical motions of the system to which is added some internal potential. This ensemble may be taken as the statistical. This enables an alternative formulation of quantum mechanics with which is associated that classical picture. Such a formulation leads, however, if developed straightforwardly according to the picture, to a theoretical scheme quite different from that of ordinary quantum mechanics concerning the problems other than the equation of motion. Furthermore, this formulation is not applicable to Fermions when spin, or exclusion principle is taken into account. Bohm's recent renewed form of the statistical interpretation of quantum mechanics referring to this formulation is criticized in several points. On this formulation we can construct a singular ensemble which is shown to correspond to the transformation kernel of the wave function to infinitesimally later instant, and thus a connection between this formulation and that of Feynman's is taken out. Hydrodynamical picture formally equivalent to the statistical picture in one-body problems is also considered, and from such analogy certain formal generalization of Schrodinger equation is suggested.