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Fock theory modification into 4-d coordinate space. Harmonic tensors in quantum Coulomb problem

Bauman Moscow State Technical University, ul. 2-ya Baumanskaya 5/1, Moscow, 105005, Russian Federation

We consider the fundamental Fock theory of hydrogen atom applied to momentum space and conformally bent by him into 3-D sphere. By such way, he created a realization of SO(3) symmetry of the Coulomb problem predicted by physicists earlier. We modify Fock theory to come to the new coordinate 4-D space from 3-D sphere by calculating 4-D Fourier transform and to arrive to Laplace equation. Tensor methods of 3-D and 4-D electrostatics are mathematically effective here to approach the Coulomb problem. Final transformation of the harmonic polinomials into the physical states is algebraic where the fourth vanishing coordinate to be equated with the radius. By the polynomial approach, we find the differential equation for states in momentum space where the integral equation known already. The quadratic Stark effect and Shwinger’s resolvent function are clearly deduced. Vector ladder operators that we found before, are considered for 3-D and 4-D harmonic tensors.

Keywords: Fock theory, quantum Coulomb problem, harmonic tensors, modification into 4-d coordinate space
PACS: 02.30.Em, 03.65.Ge, 03.65.Db (all)
DOI: 10.3367/UFNe.2021.04.038966
Citation: Efimov S P "Fock theory modification into 4-d coordinate space. Harmonic tensors in quantum Coulomb problem" Phys. Usp., accepted

Received: 3rd, April 2021, 19th, April 2021

Оригинал: Ефимов С П «Трансформация теории Фока в координатное пространство. Гармонические тензоры в квантовой задаче Кулона» УФН, принята к публикации; DOI: 10.3367/UFNr.2021.04.038966

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