On quantum multi-hamiltonian systems

On studying various examples with one degree of freedom, we discuss the possibility of extending the classical concept of multi-Hamiltonian systems to the quantum domain. We show that an adaptation is possible in each case generally leading to distinguish classes of systems characterized globally by their observable algebras.     

Quantum Hamiltonian formalism with constraints

A quantum system with constraints that does not necessarily correspond to a classical system with constraints is described in the Hamiltonian formalism.     

Unitary representations of suq(2) on the plane for q ∈ R+ or generic q ∈ S1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su _q(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions. In their study, the values of q were implicitly restricted to q R⁺. In the present paper, we extend their work to the case of generic values of q S ¹ (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q R⁺ and generic q S ¹, by determining some appropriate scalar products.     

Generalized coulomb gauges for the free photon field

We point out that the classical notion of gauge transformations can be interpreted in two different ways in the quantum theory. Beside the transformation of the field operator appearing in the Gupta-Bleuler theory, the gauge transformations can be viewed as acting on the wave functions in a generalization of the Coulomb gauge.     

The<Emphasis Type="Italic">SU</Emphasis>(1,1) coherent states related to the affine group wavelets

The Perelomov coherent states ofSU(1,1) are labeled by elements of the quotient ofSU(1,1) by its rotation subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parameterize the coherent states by elements of that group. Such a formulation permits to find new properties of theSU(1,1) coherent states and to relate them to affine wavelets. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

A three-parameter deformation of the Weyl-Heisenberg algebra: differential calculus and invariance

We define a three-parameter deformation of the Weyl-Heisenberg algebra that generalizes the q-oscillator algebra. By a purely algebraical procedure, we set up on this quantum space two differential calculi that are shown to be invariant on the same quantum group, extended to a ten-generator Hopf-star-algebra. We prove that when the values of the parameters are related, the two differential calculi reduce to one that is invariant under two quantum groups.     

Schrödinger equations for su q (2)-invariant quantum systems

By deforming the Hamiltonian of a spinless particle in a central potential we set up su _q (2)-invariant Schrödinger equations within the usual framework of quantum mechanics. Different deformations correspond to a given Hamiltonian. We explicitly solve different stationary Schrödinger equations for the free particle and for the hydrogen atom, and compare the associated energy spectra.     

su q (2)-invariant harmonic oscillator

We propose a q-deformation of the su(2)-invariant Schrödinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but also to calculate the expectation values of some physically-relevant operators. Here we consider the case of the isotropic harmonic oscillator and of the quadrupole operator governing its interaction with an external field. We obtain the spectrum and wave functions both for q R⁺ and generic q S ¹, and study the effects of the q-value range and of the arbitrariness in the su _q (2) Casimir operator choice. We then show that the quadrupole operator in l = 0 states provides a good measure of the deformation influence on the wave functions and on the Hilbert space spanned by them.     

su q (2)-invariant Schrödinger equations

We set upsu _q (2)-invariant Schrödinger equations within the usual framework of quantum mechanics. We show that the stationary equations reduce to radial ordinary differential equations by using the q-Spherical Harmonics. We apply these results to deformations of the free particle and of the hydrogen atom, for which we explicitly solve the equations and compare the deformed and undeformed theories.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

On quantum multi-Hamiltonian systems

Laboratoire de Physique Théorique et Mathématique, Université Paris VII—Denis Diderot, F. 75251 Paris Cedex 05, France. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 377–381, June, 1994.