May
28

Models of soft rotators and the theory of a harmonic rotator

Zahid Zakir [1]

Abstract

    The states of a planar oscillator are separated to a vibrational mode, containing a zero-point energy, and a rotational mode without the zero-point energy, but having a conserved angular momentum. On the basis of the analysis of properties of models of rigid and semirigid rotators, the theory of soft rotators is formulated where the harmonic attractive force is balanced only by the centrifugal force. As examples a Coulomb rotator (the Bohr model) and a magneto-harmonic rotator (the Fock-Landau levels) are considered. Disappearance of the radial speed in the model of a magneto-harmonic rotator is taken as a defining property of a pure rotational motion in the harmonic potential. After the exception of energies of the magnetic and spin decompositions, specific to magnetic fields, one turns to a simple and general model of a plane harmonic rotator (circular oscillator without radial speed) where kinetic energy is reduced to the purely rotational energy. Energy levels of the harmonic rotator have the same frequency and are twice degenerate, the energy spectrum is equidistant. In the ground state there is no zero-point energy from rotational modes, and the zero-point energy of vibrational modes can be compensated by spin effects or symmetries of the system. In this case the operators of observables vanish the ground state, i.e. are “strongly” normally ordered. In a chain of harmonic rotators collective rotations around a common axis lead to transverse waves, at quantization of which there appear quasi-particles and holes carrying an angular momentum. In the chain SU (2) appears as a group of symmetry of a rotator.

PACS: 03.65.Ge, 11.30.Er, 1130.Ly, 11.90. + t

Key words: quantization, zero-point energy, vibrations, rotations, discrete symmetries

Vol. 6, No 1, p. 1 – 13, v1,        28 May 2011

Online: TPAC: 3800-020 v2, 18 September 2012; DOI: 10.9751/TPAC.3800-020

Download  pdf 320 kb


[1] Centre for Theoretical Physics and Astrophyics, Tashkent, Uzbekistan

      zahidzakir@theor-phys.org

7 Responses to “TPAC: 3800-020 v2, Vol. 6, p. 1–13”

  1. Zahid Zakir
    October 10th, 2012 at 02:49 | #1

    1. New result:
    The states of a planar oscillator are separated to a vibrational mode, containing a zero-point energy, and a rotational mode without the zero-point energy, but having a conserved angular momentum.
    On the basis of the analysis of properties of models of rigid and semirigid rotators, the theory of soft rotators is formulated where the harmonic attractive force is balanced only by the centrifugal force.

  2. Zahid Zakir
    October 10th, 2012 at 02:50 | #2

    2. New result:
    As examples of soft rotators a Coulomb rotator (the Bohr model) and a magneto-harmonic rotator (the Fock-Landau levels) are considered.

  3. Zahid Zakir
    October 10th, 2012 at 02:53 | #3

    3. New result:
    Disappearance of the radial speed in the model of a magneto-harmonic rotator is taken as a defining property of a pure rotational motion in the harmonic potential.

  4. Zahid Zakir
    October 10th, 2012 at 02:54 | #4

    4. New result:
    After the exception of energies of the magnetic and spin decompositions, specific to magnetic fields, one turns to a simple and general model of a plane harmonic rotator (circular oscillator without radial speed) where kinetic energy is reduced to the purely rotational energy.

  5. Zahid Zakir
    October 10th, 2012 at 02:55 | #5

    5. New result:
    Energy levels of the harmonic rotator have the same frequency and are twice degenerate, the energy spectrum is equidistant. In the ground state there is no zero-point energy from rotational modes, and the zero-point energy of vibrational modes can be compensated by spin effects or symmetries of the system. In this case the operators of observables vanish the ground state, i.e. are “strongly” normally ordered.

  6. Zahid Zakir
    October 10th, 2012 at 02:58 | #6

    6. New result:
    In a chain of harmonic rotators collective rotations around a common axis lead to transverse waves, at quantization of which there appear quasi-particles and holes carrying an angular momentum.

  7. Zahid Zakir
    October 10th, 2012 at 02:58 | #7

    7. New result:
    In the chain of harmonic rotators SU (2) appears as a group of symmetry of a single rotator.

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