Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.
Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.
In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = -Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)-Hamiltonian . For an m-fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
Abstract In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)‐model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = ‐Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)‐model of the HO the three hj correspond to the self‐adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)‐Hamiltonian . For an m‐fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
The problem of degeneracy in quantum mechanics is related to the existence of groups of contact transformations under which the Hamiltonian is invariant. The correspondence between transformations in classical and quantum theories is developed. The Fock-Bargmann treatment of the symmetry group of the hydrogenic atom comes under this theory. The symmetry group of the 2-dimensional Kepler problem is found to be the 3-dimensional rotation group; that of the n-dimensional isotropic oscillator is isomorphic to the unimodular unitary group in n dimensions. The 2-dimensional anisotropic oscillator has the same symmetry as the isotropic oscillator in classical mechanics, but the quantum-mechanical problem presents complications which leave its symmetry group in doubt.
This paper addresses the general problem of zeroth order representation of resonant dynamics. We investigate the classical, quantum mechanical, and semiclassical transformation properties of two-dimensional isotropic and anisotropic uncoupled harmonic oscillators. The classical and quantal theories are presented in a manner that emphasizes the strong correspondence between the two, and in particular, the SU(2) symmetry exhibited by both the classical and quantum oscillators. The classical canonical transformations relating the action-angle variables appropriate for normal, local, and precessional motion of the isotropic oscillator are derived by explicit calculation of the generating functions. By employing a simple mapping relating the anisotropic and isotropic oscillators, expressions for action-angle variables appropriate for the topology of an arbitrary m:n resonance are determined. The resulting invariant tori are compared with the corresponding quantum mechanical wave functions and phase space densities. The relationship between the classical and quantum mechanical theories is illustrated by determining semiclassical approximations to the unitary transformation matrix elements, which are given in terms of the classical generating functions. Applications to problems of current interest, such as the adiabatic switching method for semiclassical quantization of nonseparable systems, are briefly discussed.
We contruct and study a continuous family of representations of the N-dimensional isotropic harmonic oscillator (N≥2) which are not unitarily equivalent to the standard one. We explain why such representations exist and we investigate their simpler properties: the spectrum of the Hamiltonian (which contains nonstandard values), the form of the energy eigenfunctions, and their behavior under the ladder operators. Various symmetry and dynamical groups (e.g. the rotation group) which are valid on the standard representation are not implemented on the new ones. We comment very briefly on the prospects of observing these representations experimentally.
We show that the quantum group SUq(2) can be realized in a classical symmetric-top system as long as a suitable Hamiltonian and symplectic structure are chosen. The usual Lie group can be recovered when the deformation parameter q equals 1. The Hamiltonian system in which the quantum group is realized obeys the same equations of motion with the standard Hamiltonian system, but these two Hamiltonian systems have different constants of motion that reflect that the symmetries are different.