The system of two harmonic oscillators with different sign frequencies is presented as a positive-frequency oscillator with a complex generalized coordinate where there are a global U(1) symmetry and a charge conjugation symmetry (Csymmetry).It is shown that two pairs of ladder operators, appearing at the frequency decomposition of canonical variables, are not mutually charge-conjugate and their standard interpretation as operators of charge-conjugate quanta breaks the Csymmetry.Bilinear operator identities for the ladder operators are found allowed the theory to obey the C-symmetry restrictions and to express observables through the mutually charge-conjugate operators.These identities hold for C-symmetric interactions also.It is shown that C-symmetry eliminates a ground state zero-point charge and a zero-point energy.The uncertainty relations are generalized for nonhermitian canonical variables and it is shown that the charge-conjugation symmetric ground states do not quantized.It is shown that the C-symmetric interactions do not contribute to the ground state energy in all orders of perturbation theory.
The system of two harmonic oscillators with different sign frequencies is presented as a positive-frequency oscillator with a complex generalized coordinate where there are a global U(1) symmetry and a charge conjugation symmetry (Csymmetry).It is shown that two pairs of ladder operators, appearing at the frequency decomposition of canonical variables, are not mutually charge-conjugate and their standard interpretation as operators of charge-conjugate quanta breaks the Csymmetry.Bilinear operator identities for the ladder operators are found allowed the theory to obey the C-symmetry restrictions and to express observables through the mutually charge-conjugate operators.These identities hold for C-symmetric interactions also.It is shown that C-symmetry eliminates a ground state zero-point charge and a zero-point energy.The uncertainty relations are generalized for nonhermitian canonical variables and it is shown that the charge-conjugation symmetric ground states do not quantized.It is shown that the C-symmetric interactions do not contribute to the ground state energy in all orders of perturbation theory.
The Hamiltonian of harmonic oscillator is symmetric under the replacement of canonically conjugate variables and a canonical transformation to ladder operators maintains this symmetry.The Hamiltonian is expressed through a symmetrized product of the ladder operators and, as a result, at quantization there arise a zeropoint energy.Therefore, for quantized fields, canonically conjugate variables of which enter into the Hamiltonian unsymmetrically, the zero-point energy could not arise.A new symmetry of harmonic oscillator is found: a wave equation and its solutions do not vary at a joint changing of signs of frequency, energy and mass of a particle.It is shown that the problem of the negative norm for negative-frequency states appears at taking positive mass at negative energy and, contrary, the problem disappears at taking the same sign of mass and energy as it is required by relativistic kinematics.In the nonrelativistic theory, considered as a limiting case of relativistic theory, the states of a particle with negative frequency, energy and mass are described consistently as evolving only backward in time and representing the states of its antiparticle with positive frequency, energy and mass evolving forward in time.For such charge-conjugation symmetric system of oscillators an extended space of states with generalized operators is constructed.
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 zeropoint 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.
Arbitrarily small changes in the commutation relations suffice to transform the usual singular quantum theories into regular quantum theories. This process is an extension of canonical quantization that we call general quantization. Here we apply general quantization to the time-independent linear harmonic oscillator. The unstable Heisenberg group becomes the stable group SO(3). This freezes out the zero-point energy of very soft or very hard oscillators, like those responsible for the infrared or ultraviolet divergencies of usual field theories, without much changing the medium oscillators. It produces pronounced violations of equipartition and of the usual uncertainty relations for soft or hard oscillators, and interactions between the previously uncoupled excitation quanta of the oscillator, weakly attractive for medium quanta, strongly repulsive for soft or hard quanta.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
Following the ideas of Wigner, we quantize noncanonically a system of two nonrelativistic point particles, interacting via a harmonic potential. The center of mass phase-space variables are quantized in a canonical way, whereas the internal momentum and coordinates are assumed to satisfy relations, which are essentially different from the canonical commutation relations. As a result, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. In particular, the distance between the constituents is preserved in time and can take at most four different values. The orbital momentum is either zero or one (in units ℏ/2). The operators of the coordinates do not commute with each other and, therefore, the position of any one of the constituents cannot be localized; the particles are smeared with a certain probability in a finite space volume, which moves together with the center of mass. In the limit ℏ→0 the constituents ‘‘fall’’ into their center of mass and the composite system behaves as a free point particle.
\n We have formulated a model of a complex (two-dimensional) quantum harmonic oscillator. All dynamical physical variables are expressed in terms of the creation and annihilation operators, viz., $a_{z}, \\bar{a}_{z}\\ {\\rm and} \\ a_{\\bar{z}}, \\bar{a}_{\\bar{z}} $. The Hamiltonian of the system is $H_{z\\bar{z}} = (\\bar{a}_{z}a_{z} + 1) + \\omega L_{z} $, where ω is the oscillator frequency and $L_{z} = \\frac{\\hbar }{2}\\,(\\bar{a}_{\\bar{z}} a_{\\bar{z}}-\\bar{a}_{z}a_{z}) $ is the orbital angular momentum. The oscillator is found to be described by a conserved orbital angular momentum (Lz) besides energy. While the ground-state wave function is real, all excited states are complex and degenerate. The oscillator in these states carry a quantum of charge of $e^{{\\ast}} = \\frac{n}{2n\\pm1} e $. These degenerate wave functions are eigenstates of the orbital angular momentum with eigenvalues nℏ and −nℏ, where h=2πℏ is the Planck's constant and n=1, 2, … . The two wave functions are degenerate with energy En=(n+1)ℏω. The comparison with Landau level reveals that in the presence of the magnetic field, B, where ω is equal to the cyclotron frequency, the current moment is quantized and is proportional to the square root of the magnetic field, i.e., ${\\cal I}_{n}\\propto n e \\sqrt{B} $.\n