Abstract
As a generalization and extension of our previous paper [Turbiner et al., J. Phys. A: Math. Theor. 53, 055302 (2020)], in this work, we study a quantum four-body system in Rd (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, rij ≡ |ri − rj|, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ρij≡rij2, the corresponding reduced Hamiltonian of the system possesses a hidden sl(7; R) Lie algebra structure. In the ρ-representation, it is shown that the four-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly solvable. We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite and three others are equal), molecular two-center (two masses are infinite and two others are equal), and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born–Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. In addition, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for the four-body harmonic oscillator case, there exists an infinite family of eigenfunctions that depend on the single variable, which is the moment of inertia of the system.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.