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arXiv:2411.14482

Runge-Lenz operator in the momentum space

Sergei P. Efimov

Published Nov 19, 2024• 0 citations
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Abstract

The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) simmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and the Runge-Lenz operators.This approach is unknown in the momentum space where the Runge-Lenz operator is not applied in the existing theory. The Runge-Lenz operator obtained in the momentum space is simplier than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional the Fock sphere has been determined.

DOI
10.48550/arxiv.2411.14482

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The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) simmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and the Runge-Lenz operators.This approach is unknown in the momentum space where the Runge-Lenz operator is not applied in the existing theory. The Runge-Lenz operator obtained in the momentum space is simplier than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional the Fock sphere has been determined.
The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) symmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and Runge–Lenz operators. This approach is unknown in the momentum space where the Runge–Lenz operator is not applied. The Runge–Lenz operator obtained in the momentum space is simpler than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional Fock sphere has been determined.
The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) symmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and Runge–Lenz operators. This approach is unknown in the momentum space where the Runge–Lenz operator is not applied. The Runge–Lenz operator obtained in the momentum space is simpler than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional Fock sphere has been determined.
We consider Fock's fundamental theory of the hydrogen atom in momentum space which allows a realization of the previously predicted rotation group of a three-dimensional (3D) sphere in four-dimensional (4D) space. We then modify Fock's theory and abandon the momentum space description. To transform and simplify the theory, we use invariant tensor methods of electrostatics in 3D and 4D spaces. We find a coordinate 4D space where the Schrodinger equation becomes the 4D Laplace equation. The transition from harmonic 4D polynomials to original 3D physical space is algebraic and involves derivatives with respect to a coordinate that is interpreted as time. We obtain a differential equation for eigenfunctions in the momentum space and find its solutions. A concise calculation of the quadratic Stark effect is given. The Schwinger resolvent is derived by the method of harmonic polynomials. Vector ladder operators are also considered.
A new mathematical method is established to represent the operator, wave functions and square matrix in the same representation. We can obtain the specific square matrices corresponding to the angular momentum and Runge-Lenz vector operators with invoking assistance from the operator relations and the orthonormal wave functions. Furthermore, the first-order differential equations will be given to deduce the specific wave functions without using the solution of the second order Schrodinger equation. As a result, we will unify the descriptions of the matrix mechanics and the wave mechanics on hydrogen atom. By using matrix transformations, we will also deduce the specific matrix representations of the operators in the SO (4,2) group.
We have constructed a set of non-Hermitian operators that satisfy the commutation relations of the SO(3)-Lie algebra. It is shown that this operators generate rotations in the configuration space and not in the momentum space but in a modified non-Hermitian momentum space. This generators are related with a new type of spherical harmonics that result to be PT-orthonormal. Additionally, we have shown that this operators represent conserved quantities for a non-Hermitian Hamiltonian with an additional complex term. As a particular case, the solutions of the corresponding PT-Hydrogen atom that includes a complex term are obtained, and it is found that a non-Hermitian Runge-Lenz vector is a conserved quantity. In this way, we obtain a set of non-Hermitian operators that satisfy the SO(4)-Lie algebra.