Models of soft rotators and the theory of a harmonic rotator

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 planar 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.