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

arXiv:1111.5569

On The Harmonic Oscillator Group

Raquel M. López, Sergeĭ K. Suslov, José M. Vega-Guzmán

Published Nov 23, 2011• 0 citations
arXivHTMLPDF

Abstract

We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a view point of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schroedinger group of the free particle.

DOI
10.48550/arxiv.1111.5569

Related papers

On a hidden symmetry of quantum harmonic oscillators

Raquel M. López, Sergeĭ K. Suslov, José M. Vega-Guzmán

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.

The Journal of Difference Equations and Applications•Feb 21, 2012•30 citations

The maximal kinematical invariance group of the harmonic oscillator

U. Niederer

The largest group of coordinate transformations leaving invariant the Schroedinger equation of the n-dimensional harmonic oscillator is determined and shown to be isomorphic to the corresponding group of the free-particle equation. It can be described as a Galilei group in which the time translations have been replaced by the group SL(2,R) of projective transformations. The relation between the oscillator group and the spectrumgenenating algebra of the harmonic oscillator is investigated. The relevance of the oscillator group and the group SL(2,R) for general quantum systems is discussed. (auth)

Helvetica Physica Acta (Switzerland)•Aug 15, 1973•166 citations

On a Hidden Symmetry of Quantum Harmonic Oscillators

Raquel M. López, Sergeĭ K. Suslov, José M. Vega-Guzmán

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.

arXiv:1112.2586•Dec 12, 2011•1 citation

An analogue of the Berry phase for simple harmonic oscillators

Sergeĭ K. Suslov

We evaluate a variant of Berry's phase for a ?missing? family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical?invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found here by integration with the help of a computer algebra system.

Physica Scripta•Feb 11, 2013•8 citations

The Berry Phase for Simple Harmonic Oscillators

Sergeĭ K. Suslov

We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system.

arXiv:1112.2418•Dec 12, 2011•4 citations

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

M. Fernández‐Guasti, H. M. Moya-Cessa

The correspondence between classical and quantum invariants is established. The Ermakov-Lewis quantum invariant of the time-dependent harmonic oscillator is translated from the coordinate and momentum operators into amplitude and phase operators. In doing so, Turski's phase operator as well as Susskind-Glogower operators are generalized to the time-dependent harmonic-oscillator case. A quantum derivation of the Manley-Rowe relations is shown as an example.

arXiv:quant-ph/0212073•Physical Review A•Jun 11, 2003•28 citations
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a view point of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schroedinger group of the free particle.
We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.
The largest group of coordinate transformations leaving invariant the Schroedinger equation of the n-dimensional harmonic oscillator is determined and shown to be isomorphic to the corresponding group of the free-particle equation. It can be described as a Galilei group in which the time translations have been replaced by the group SL(2,R) of projective transformations. The relation between the oscillator group and the spectrumgenenating algebra of the harmonic oscillator is investigated. The relevance of the oscillator group and the group SL(2,R) for general quantum systems is discussed. (auth)
We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.