Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators
Dirac’s Generalized Hamiltonian Dynamics (GHD) is a purely classical formalism for systems having constraints: it incorporates the constraints into the Hamiltonian. Dirac designed the GHD specifically for applications to quantum field theory. In one of our previous papers, we redesigned Dirac’s GHD...
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MDPI AG
2020-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/7/1130 |
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| author | Eugene Oks |
| author_facet | Eugene Oks |
| author_sort | Eugene Oks |
| collection | DOAJ |
| description | Dirac’s Generalized Hamiltonian Dynamics (GHD) is a purely classical formalism for systems having constraints: it incorporates the constraints into the Hamiltonian. Dirac designed the GHD specifically for applications to quantum field theory. In one of our previous papers, we redesigned Dirac’s GHD for its applications to atomic and molecular physics by choosing integrals of the motion as the constraints. In that paper, after a general description of our formalism, we considered hydrogenic atoms as an example. We showed that this formalism leads to the existence of classical non-radiating (stationary) states and that there is an infinite number of such states—just as in the corresponding quantum solution. In the present paper, we extend the applications of the GHD to a charged Spherical Harmonic Oscillator (SHO). We demonstrate that, by using the higher-than-geometrical symmetry (i.e., the algebraic symmetry) of the SHO and the corresponding additional conserved quantities, it is possible to obtain the classical non-radiating (stationary) states of the SHO and that, generally speaking, there is an infinite number of such states of the SHO. Both the existence of the classical stationary states of the SHO and the infinite number of such states are consistent with the corresponding quantum results. We obtain these new results from first principles. Physically, the existence of the classical stationary states is the manifestation of a non-Einsteinian time dilation. Time dilates more and more as the energy of the system becomes closer and closer to the energy of the classical non-radiating state. We emphasize that the SHO and hydrogenic atoms are not the only microscopic systems that can be successfully treated by the GHD. All classical systems of N degrees of freedom have the algebraic symmetries O<sub>N+1</sub> and SU<sub>N</sub>, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely O<sub>4</sub> and SU<sub>3</sub>; they possess an additional vector integral of the motion, while the quantal counterpart-operator does not exist. This offers possibilities that are absent in quantum mechanics. |
| first_indexed | 2024-03-10T18:38:58Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T18:38:58Z |
| publishDate | 2020-07-01 |
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| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-dd4dc32e554b44cd8474d6e010c6f4002023-11-20T06:02:34ZengMDPI AGSymmetry2073-89942020-07-01127113010.3390/sym12071130Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic OscillatorsEugene Oks0Physics Department, Auburn University, 380 Duncan Drive, Auburn, AL 36849, USADirac’s Generalized Hamiltonian Dynamics (GHD) is a purely classical formalism for systems having constraints: it incorporates the constraints into the Hamiltonian. Dirac designed the GHD specifically for applications to quantum field theory. In one of our previous papers, we redesigned Dirac’s GHD for its applications to atomic and molecular physics by choosing integrals of the motion as the constraints. In that paper, after a general description of our formalism, we considered hydrogenic atoms as an example. We showed that this formalism leads to the existence of classical non-radiating (stationary) states and that there is an infinite number of such states—just as in the corresponding quantum solution. In the present paper, we extend the applications of the GHD to a charged Spherical Harmonic Oscillator (SHO). We demonstrate that, by using the higher-than-geometrical symmetry (i.e., the algebraic symmetry) of the SHO and the corresponding additional conserved quantities, it is possible to obtain the classical non-radiating (stationary) states of the SHO and that, generally speaking, there is an infinite number of such states of the SHO. Both the existence of the classical stationary states of the SHO and the infinite number of such states are consistent with the corresponding quantum results. We obtain these new results from first principles. Physically, the existence of the classical stationary states is the manifestation of a non-Einsteinian time dilation. Time dilates more and more as the energy of the system becomes closer and closer to the energy of the classical non-radiating state. We emphasize that the SHO and hydrogenic atoms are not the only microscopic systems that can be successfully treated by the GHD. All classical systems of N degrees of freedom have the algebraic symmetries O<sub>N+1</sub> and SU<sub>N</sub>, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely O<sub>4</sub> and SU<sub>3</sub>; they possess an additional vector integral of the motion, while the quantal counterpart-operator does not exist. This offers possibilities that are absent in quantum mechanics.https://www.mdpi.com/2073-8994/12/7/1130generalized Hamiltonian dynamicsspherical harmonic oscillatorclassical non-radiating stationary statesalgebraic symmetry of classical systems |
| spellingShingle | Eugene Oks Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators Symmetry generalized Hamiltonian dynamics spherical harmonic oscillator classical non-radiating stationary states algebraic symmetry of classical systems |
| title | Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators |
| title_full | Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators |
| title_fullStr | Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators |
| title_full_unstemmed | Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators |
| title_short | Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators |
| title_sort | application of the generalized hamiltonian dynamics to spherical harmonic oscillators |
| topic | generalized Hamiltonian dynamics spherical harmonic oscillator classical non-radiating stationary states algebraic symmetry of classical systems |
| url | https://www.mdpi.com/2073-8994/12/7/1130 |
| work_keys_str_mv | AT eugeneoks applicationofthegeneralizedhamiltoniandynamicstosphericalharmonicoscillators |