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

American Journal of Physics

On the isotropic oscillator and the hydrogenic atom in classical and quantum mechanics

Ronald M. Rockmore

Published Jan 1, 1975• 12 citations
Publisher

Abstract

It is shown how one may simply associate the problem of the isotropic oscillator to that of the hydrogenic atom in classical dynamics, particularly in its action−angle variable formulation, so that the solution of the one problem implies that of the other. This relationship persists in the two−dimensional quantum mechanics and provides the key to the construction of a wave pocket solution for the isotropic oscillator in the region of large principal quantum number in three dimensions.

DOI
10.1119/1.10010

Related papers

The relations between d-dimensional isotropic oscillator and D-dimensional like-hydrogen atom

Zahra Bakhshi, Zahra Neshati

Being comparable in quantum systems makes it possible for spaces with varying dimensions to attribute each other using special conversions can attribute schrodinger equation with like-hydrogen atom potential in defined dimensions to a schrodinger equation with other certified dimensions with isotropic oscillator potential. Applying special transformation provides a relationship between different dimensions of two quantum systems. The result of the quantized isotropic oscillator can be generalized to like-hydrogen atom problem in different dimensions. The connection between coordinate spaces in different dimensions can follow a specific relation that by using it and applying the parametric definition in two problems, energy spectrum and like-hydrogen atom potential wave functions problem will be solved.

arXiv:1904.10830•Apr 24, 2019•0 citations

On the Problem of Degeneracy in Quantum Mechanics

J. M. Jauch, Edward L. Hill

The problem of degeneracy in quantum mechanics is related to the existence of groups of contact transformations under which the Hamiltonian is invariant. The correspondence between transformations in classical and quantum theories is developed. The Fock-Bargmann treatment of the symmetry group of the hydrogenic atom comes under this theory. The symmetry group of the 2-dimensional Kepler problem is found to be the 3-dimensional rotation group; that of the $n$-dimensional isotropic oscillator is isomorphic to the unimodular unitary group in $n$ dimensions. The 2-dimensional anisotropic oscillator has the same symmetry as the isotropic oscillator in classical mechanics, but the quantum-mechanical problem presents complications which leave its symmetry group in doubt.

Physical Review•Apr 1, 1940•369 citations

Classical, quantum mechanical, and semiclassical representations of resonant dynamics: A unified treatment

Craig C. Martens, Gregory S. Ezra

This paper addresses the general problem of zeroth order representation of resonant dynamics. We investigate the classical, quantum mechanical, and semiclassical transformation properties of two-dimensional isotropic and anisotropic uncoupled harmonic oscillators. The classical and quantal theories are presented in a manner that emphasizes the strong correspondence between the two, and in particular, the SU(2) symmetry exhibited by both the classical and quantum oscillators. The classical canonical transformations relating the action-angle variables appropriate for normal, local, and precessional motion of the isotropic oscillator are derived by explicit calculation of the generating functions. By employing a simple mapping relating the anisotropic and isotropic oscillators, expressions for action-angle variables appropriate for the topology of an arbitrary m:n resonance are determined. The resulting invariant tori are compared with the corresponding quantum mechanical wave functions and phase space densities. The relationship between the classical and quantum mechanical theories is illustrated by determining semiclassical approximations to the unitary transformation matrix elements, which are given in terms of the classical generating functions. Applications to problems of current interest, such as the adiabatic switching method for semiclassical quantization of nonseparable systems, are briefly discussed.

The Journal of Chemical Physics•Jul 1, 1987•43 citations

Homogeneous canonical formulation of the nonrelativistic hydrogen atom

Horacio Grinberg, Julio Marañón, H. Vucetich

The homogeneous canonical formulation is applied to the Hamiltonian of the nonrelativistic hydrogen atom. Its connection with the isotropic harmonic oscillator in a four-dimensional Riemann space leads to the quantum analog of the Kepler problem.

Journal of Mathematical Physics•Sep 1, 1984•5 citations

Classical limit of the two-dimensional and the three-dimensional hydrogen atom

S Nandi, C. S. Shastry

Using the harmonic oscillator representation of the hydrogen atom and constructing the appropriate coherent state corresponding to the minimum uncertainty product, the classical limits of the two-dimensional and the three-dimensional hydrogen atom are examined. The method adopted is similar to that used by Bhaumik, Dutta Roy and Ghosh (1986). The authors deduce the classical limit by requiring that the expectation value (r) of the radial variable is large for the two-dimensional hydrogen atom. The resulting trajectory is an ellipse satisfying the conditions of the Kepler orbit in classical mechanics. In order to obtain the classical limit of the three-dimensional hydrogen atom, besides the condition of large (r), they make the uncertainty Delta r=((r 2 )-(r) 2 ) 1/2 a minimum with respect to certain parameters of the coherent state. Imposition of only the first condition leads to an elliptical orbit, which is not a Kepler orbit in general. Imposition of both conditions leads to an orbit identical with that of the two-dimensional hydrogen atom. However, the time evolution of the expectation values of position variables are consistent with the corresponding results of the Kepler problem only for small values of eccentricity, but not in general.

Journal of Physics A Mathematical and General•Apr 21, 1989•14 citations

Hydrogen atom in the phase-space formulation of quantum mechanics

José M. Gracia-Bondı́a

Using a coordinate transformation which regularizes the classical Kepler problem, we show that the hydrogen-atom case may be analytically solved via the phase-space formulation of nonrelativistic quantum mechanics. The problem is essentially reduced to that of a four-dimensional oscillator whose treatment in the phase-space formulation is developed. Furthermore, the method allows us to calculate the Green's function for the H atom in a surprisingly simple way.

Physical Review A•Aug 1, 1984•59 citations
It is shown how one may simply associate the problem of the isotropic oscillator to that of the hydrogenic atom in classical dynamics, particularly in its action−angle variable formulation, so that the solution of the one problem implies that of the other. This relationship persists in the two−dimensional quantum mechanics and provides the key to the construction of a wave pocket solution for the isotropic oscillator in the region of large principal quantum number in three dimensions.
Being comparable in quantum systems makes it possible for spaces with varying dimensions to attribute each other using special conversions can attribute schrodinger equation with like-hydrogen atom potential in defined dimensions to a schrodinger equation with other certified dimensions with isotropic oscillator potential. Applying special transformation provides a relationship between different dimensions of two quantum systems. The result of the quantized isotropic oscillator can be generalized to like-hydrogen atom problem in different dimensions. The connection between coordinate spaces in different dimensions can follow a specific relation that by using it and applying the parametric definition in two problems, energy spectrum and like-hydrogen atom potential wave functions problem will be solved.
The problem of degeneracy in quantum mechanics is related to the existence of groups of contact transformations under which the Hamiltonian is invariant. The correspondence between transformations in classical and quantum theories is developed. The Fock-Bargmann treatment of the symmetry group of the hydrogenic atom comes under this theory. The symmetry group of the 2-dimensional Kepler problem is found to be the 3-dimensional rotation group; that of the nnn-dimensional isotropic oscillator is isomorphic to the unimodular unitary group in nnn dimensions. The 2-dimensional anisotropic oscillator has the same symmetry as the isotropic oscillator in classical mechanics, but the quantum-mechanical problem presents complications which leave its symmetry group in doubt.
This paper addresses the general problem of zeroth order representation of resonant dynamics. We investigate the classical, quantum mechanical, and semiclassical transformation properties of two-dimensional isotropic and anisotropic uncoupled harmonic oscillators. The classical and quantal theories are presented in a manner that emphasizes the strong correspondence between the two, and in particular, the SU(2) symmetry exhibited by both the classical and quantum oscillators. The classical canonical transformations relating the action-angle variables appropriate for normal, local, and precessional motion of the isotropic oscillator are derived by explicit calculation of the generating functions. By employing a simple mapping relating the anisotropic and isotropic oscillators, expressions for action-angle variables appropriate for the topology of an arbitrary m:n resonance are determined. The resulting invariant tori are compared with the corresponding quantum mechanical wave functions and phase space densities. The relationship between the classical and quantum mechanical theories is illustrated by determining semiclassical approximations to the unitary transformation matrix elements, which are given in terms of the classical generating functions. Applications to problems of current interest, such as the adiabatic switching method for semiclassical quantization of nonseparable systems, are briefly discussed.
Using the harmonic oscillator representation of the hydrogen atom and constructing the appropriate coherent state corresponding to the minimum uncertainty product, the classical limits of the two-dimensional and the three-dimensional hydrogen atom are examined. The method adopted is similar to that used by Bhaumik, Dutta Roy and Ghosh (1986). The authors deduce the classical limit by requiring that the expectation value (r) of the radial variable is large for the two-dimensional hydrogen atom. The resulting trajectory is an ellipse satisfying the conditions of the Kepler orbit in classical mechanics. In order to obtain the classical limit of the three-dimensional hydrogen atom, besides the condition of large (r), they make the uncertainty Delta r=((r 2 )-(r) 2 ) 1/2 a minimum with respect to certain parameters of the coherent state. Imposition of only the first condition leads to an elliptical orbit, which is not a Kepler orbit in general. Imposition of both conditions leads to an orbit identical with that of the two-dimensional hydrogen atom. However, the time evolution of the expectation values of position variables are consistent with the corresponding results of the Kepler problem only for small values of eccentricity, but not in general.