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Journal of Mathematical Physics

Coherent distributions of the symmetry algebra of spinor regularization generator and hydrogen atom

E. M. Novikova

Published Aug 1, 2024• 2 citations
Publisher

Abstract

A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.

DOI
10.1063/5.0172309

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A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.
An unusual family of generalized coherent states of the Heisenberg algebra is constructed. It consists of two families of functionals on the Schwartz space. Each of them has the key properties of ordinary coherent states: it intertwines the representations of a given algebra, has the completeness property, and minimizes the product of variances in the Heisenberg relation. However, unlike the usual coherent states that are eigenstates for the annihilation operator, the constructed distributions belong to the continuous spectrum of Hermitian generators of the Heisenberg algebra. The inner product of functionals from different families has the main properties of the overlap function of ordinary coherent states: it is continuous in parameters, satisfies the reproducing identity, and has the corresponding geometric meaning. The application of unusual coherent states of the continuous spectrum is demonstrated by an example of solving the spectral problem for an inverted oscillator.
We construct the systems of generalised coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in terms of elementary functions and are invariant under the SO(3,2)SO(3, 2)SO(3,2) (discrete spectrum) and SO(4,1)SO(4, 1)SO(4,1) (continuous spectrum) subgroups of the dynamical symmetry group SO(4,2)SO(4, 2)SO(4,2) of the hydrogen atom. Both systems of coherent states are particular cases of the kernel of integral operator which interwines irreducible representations of the SO(4,2)SO(4, 2)SO(4,2) group.
We propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start examining a pioneering work, by Gazeau and Klauder, where the concept of coherent states for systems with discrete spectrum was extended to systems with continuous one. In the present article, we see the Husimi distribution as a representation of the density operator in terms of a basis of coherent states. There are other ways to obtain it, but we do not consider here. We specially discuss the problem of the continuous harmonic oscillator.
In this paper we start to construct some fundamental features of Dirac Calculus, specifically, we go inside the theory of Heisenberg continuous matrices, which, in our Schwartz Linear Algebra, are represented by Schwartz families. We distinguish the important subclass of transposable continuous matrices and give some basic and very important examples in Quantum Mechanics. So we define transposable Schwartz families and their transpose families, we prove the transposability of Dirac families and Fourier families. We find the transpose of regular-distribution families in a much general case. We define symmetric families, the analogous of symmetric ma- trices in the continuous case. We prove the symmetry of Dirac families and of Fourier families. We define Hermitian families, the analogous of Hermitian matrices in the continuous case. We prove the Hermitianity of Dirac families and of Fourier families. We define unitary families, the analogous of unitary matrices in the continuous case. We prove the unitarity of Dirac families and of the fundamental normalized de Broglie family. Then, we use the transpose of a family to find the components of the superpositions of transposable families, we give a general result and we apply this result to the Dirac families and the eigenfamilies of the vector-wave operator. We shall use the transposable families in next chapters to define the Dirac product in distribution spaces, basic product for the entire foundation of Dirac Calculus and Quantum Mechanics formalism.