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Journal of Physics A Mathematical and General

On parasupersymmetric Hamiltonians and vector mesons in magnetic fields

N. Debergh

Published Apr 7, 1994• 7 citations
Publisher

Abstract

A new Sakata-Taketani equation describing vector mesons in interaction with a constant magnetic field is proposed. It leads to real energies as opposed to relativistic equations, which have already been pointed out, associated with this particular interacting context. In particular, the connection between this new proposal and a specific model coming from parasupersymmetric quantum mechanics is studied.

DOI
10.1088/0305-4470/27/7/004

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A new Sakata-Taketani equation describing vector mesons in interaction with a constant magnetic field is proposed. It leads to real energies as opposed to relativistic equations, which have already been pointed out, associated with this particular interacting context. In particular, the connection between this new proposal and a specific model coming from parasupersymmetric quantum mechanics is studied.
Results coming from the study of relativistic vector mesons interacting with a constant magnetic field are examined through Johnson-Lippmann implications on one-dimensional oscillatorlike systems. We obtain specific nonrelativistic Hamiltonians showing new properties in quantum mechanics and leading to superpositions of bosons and pseudofermions. Moreover, two “potentials” are introduced and discussed in comparison with recent developments usually obtained in p=2 parasupersymmetric quantum mechanics. Pseudofermions are also examined, particularly with respect to orthofermions.
The motion of a charged particle in a constant magnetic field is treated in both relativistic and non-relativistic quantum theory. Operators representing the center of the orbit, which obey the commutation law for conjugate variables, are introduced and their connections with energy, angular momentum, and magnetic moment studied. Energy eigenfunctions in an operator form are obtained by factorization. Previously derived eigenfunctions in coordinate space are obtained and are shown to be eigenfunctions for the operators for the center of the orbit as well as for the energy. Corresponding relativistic eigenfunctions are derived by a simple device which enables one to construct solutions of the Dirac equation from solutions of the Schr\"odinger equation.
Pseudosupersymmetric quantum mechanics (PsSSQM), based upon the use of pseudofermions, was introduced in the context of a new Kemmer equation describing charged vector mesons interacting with an external constant magnetic field. Here we construct the complete explicit solution for its realization in terms of two superpotentials, both equal or unequal. We prove that any orthosupersymmetric quantum mechanical system has a pseudosupersymmetry and give conditions under which a pseudosupersymmetric one may be described by orthosupersymmetries of order two. We propose two new matrix realizations of PsSSQM in terms of the generators of a generalized deformed oscillator algebra (GDOA) and relate them to the cases of equal or unequal superpotentials, respectively. We demonstrate that these matrix realizations are fully reducible and that their irreducible components provide two distinct sets of bosonized operators realizing PsSSQM and corresponding to nonlinear spectra. We relate such results to some previous ones obtained for a GDOA connected with a C3-extended oscillator algebra (where C3 = ℤ3) in the case of linear spectra.
Up to this point in our study of quantum mechanics, we have ignored relativistic effects and taken spin as an extra observed property of particles with no specific model. In formulating an appropriate theory for the extension of quantum mechanics to include relativistic effects, it is logical to begin with the Hamiltonian based on our knowledge of relativity, where we have Hˆ = c[(pˆ− eA)2 +m2c2]1/2 + eφ , (11.1) where the first portion portion comes from W 2 = p2c2 + m2c4 using the canonical momentum pˆ = γmv + eA when there is a magnetic field present represented by the vector potential A, and then we have added the scalar electrostatic potential. This form for the Hamiltonian is very formidable, since it includes a square root with an operator inside (pˆ → (~/i)∇). By rearranging and squaring, Equation (11.1) can be put into a more symmetric form that avoids this difficulty, so that (Hˆ − eφ)2 − (pˆ− eA)2c2 = m2c4 . (11.2) One possible interpretation of this equation is that we should make the usual substitutions pˆ→ ~ i ∇ and Hˆ → −~ i ∂ ∂t , which leads to[( −~ i ∂ ∂t − eφ )2 − ( ~ i ∇− eA )2 c2 ] ψ = m2c4ψ . (11.3) This equation is called the Klein-Gordon equation, and represents one possible way to generalize the Schro¨dinger equation to include relativistic effects.