Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Two quantum extra symmetries and associated potentials

R. P. Martı́nez-y-Romero, H. N. Núñez‐Yépez, A. L. Salas‐Brito

Published Mar 1, 2026• 0 citations
Publisher

Abstract

When a non-relativistic quantum particle moving under a radial potential V(r) has either an extra conserved vectorial operator or an extra conserved tensorial operator then, in the first case, the corresponding operator must be proportional to the quantum Laplace-Runge-Lenz vector and the compatible potential has to be the Coulomb potential. While, in the second case, the compatible potential is the harmonic oscillator one and the conserved quantity has to be a tensor formed by a symmetric combination of the position and the momentum operators, which is the quantum version of the classically conserved tensor. Thus, the superintegrable condition on classical 3D radial problems producing only closed, periodic and plane orbits still manages, even in the quantum case, to impose strong restrictions on the interaction potentials.

DOI
10.1063/5.0241515

Related papers

The classical Lenz vector and the two-dimensional quantum harmonic oscillator

Stefano Palmisano, Antonio Davide Polosa, R. Senese +1 more

Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.

Physics Open•Apr 8, 2020•0 citations

Three-Dimensional Isotropic Harmonic Oscillator and SU3

David Milton Fradkin

A consideration of the eigenvalue problem for the quantum-mechanical three-dimensional isotropic harmonic oscillator leads to a derivation of a conserved symmetric tensor operator, in addition to the angular momentum vector operator. Upon examining the classical limit, it is found that the symmetric tensor completely specifies the orientation of the elliptical orbit, in a way analogous to the Runge-Lenz vector for the Kepler problem. The tensor operator and the angular momentum vector operator are then related to the infinitesimal operators for the SU3 group.

American Journal of Physics•Mar 1, 1965•175 citations

Motion in Centrally Symmetric Potentials

David K. Ferry

The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. This simple problem of a particle in a quadratic potential energy is one of the typical problems of quantization. In two dimensions, the problem is more interesting, but not particularly more complicated. In three dimensions, people could extend the treatment to a three-dimensional harmonic oscillator, but a more important problem has spherical symmetry in the potential, and the latter is not the quadratic potential of the harmonic oscillator. It is of more interest to turn our attention to the motion of an electron about the nucleus of an atom.

CRC Press eBooks•Nov 6, 2020•0 citations

An integrated approach to ladder and shift operators for the Morse oscillator, radial Coulomb and radial oscillator potentials

Ian L. Cooper

The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.

Journal of Physics A Mathematical and General•Apr 7, 1993•79 citations

On the connection between the radial momentum operator and the Hamiltonian in n dimensions

Gil Paz

The radial momentum operator in quantum mechanics is usually obtained through canonical quantization of the (symmetrical form of the) classical radial momentum. We show that the well known connection between the Hamiltonian of a free particle and the radial momentum operator $\hat{H}=\hat{P}_{r}^2/2m+ $\hat{L}^2$}/2mr^{2}$ is true only in one or three dimensions. In general, an extra term of the form $\hbar^{2}(n-1)(n-3)/ 2m \cdot 4r^{2}$ has to be added to the Hamiltonian.

arXiv:quant-ph/0009046•European Journal of Physics•Sep 11, 2000•24 citations

Non-Hermitian Radial Momentum Operator and Path Integrals in Polar Coordinates

Kazuo Fujikawa

A salient feature of the Schrödinger equation is that the classical radial momentum term p2r in polar coordinates is replaced by the operator , where the operator is not hermitian in general. This fact has important implications for the path integral and semi-classical approximations. When one defines a formal hermitian radial momentum operator , the relation holds in d-dimensional space and this extra potential appears in the path integral formulated in polar coordinates. The extra potential, which influences the classical solutions in the semi-classical treatment such as in the analysis of solitons and collective modes, vanishes for d = 3 and attractive for d = 2 and repulsive for all other cases d ≥ 4. This extra term induced by the non-hermitian operator is a purely quantum effect, and it is somewhat analogous to the quantum anomaly in chiral gauge theory.

arXiv:0805.3879•Progress of Theoretical Physics•Aug 1, 2008•9 citations
When a non-relativistic quantum particle moving under a radial potential V(r) has either an extra conserved vectorial operator or an extra conserved tensorial operator then, in the first case, the corresponding operator must be proportional to the quantum Laplace-Runge-Lenz vector and the compatible potential has to be the Coulomb potential. While, in the second case, the compatible potential is the harmonic oscillator one and the conserved quantity has to be a tensor formed by a symmetric combination of the position and the momentum operators, which is the quantum version of the classically conserved tensor. Thus, the superintegrable condition on classical 3D radial problems producing only closed, periodic and plane orbits still manages, even in the quantum case, to impose strong restrictions on the interaction potentials.
Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.
A consideration of the eigenvalue problem for the quantum-mechanical three-dimensional isotropic harmonic oscillator leads to a derivation of a conserved symmetric tensor operator, in addition to the angular momentum vector operator. Upon examining the classical limit, it is found that the symmetric tensor completely specifies the orientation of the elliptical orbit, in a way analogous to the Runge-Lenz vector for the Kepler problem. The tensor operator and the angular momentum vector operator are then related to the infinitesimal operators for the SU3 group.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. This simple problem of a particle in a quadratic potential energy is one of the typical problems of quantization. In two dimensions, the problem is more interesting, but not particularly more complicated. In three dimensions, people could extend the treatment to a three-dimensional harmonic oscillator, but a more important problem has spherical symmetry in the potential, and the latter is not the quadratic potential of the harmonic oscillator. It is of more interest to turn our attention to the motion of an electron about the nucleus of an atom.
The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.
The radial momentum operator in quantum mechanics is usually obtained through canonical quantization of the (symmetrical form of the) classical radial momentum. We show that the well known connection between the Hamiltonian of a free particle and the radial momentum operator $\hat{H}=\hat{P}_{r}^2/2m+ L^2\hat{L}^2L^2}/2mr^{2}$ is true only in one or three dimensions. In general, an extra term of the form ℏ2(n−1)(n−3)/2m⋅4r2\hbar^{2}(n-1)(n-3)/ 2m \cdot 4r^{2}ℏ2(n−1)(n−3)/2m⋅4r2 has to be added to the Hamiltonian.
A salient feature of the Schrödinger equation is that the classical radial momentum term p2r in polar coordinates is replaced by the operator , where the operator is not hermitian in general. This fact has important implications for the path integral and semi-classical approximations. When one defines a formal hermitian radial momentum operator , the relation holds in d-dimensional space and this extra potential appears in the path integral formulated in polar coordinates. The extra potential, which influences the classical solutions in the semi-classical treatment such as in the analysis of solitons and collective modes, vanishes for d = 3 and attractive for d = 2 and repulsive for all other cases d ≥ 4. This extra term induced by the non-hermitian operator is a purely quantum effect, and it is somewhat analogous to the quantum anomaly in chiral gauge theory.