The Experts below are selected from a list of 315 Experts worldwide ranked by ideXlab platform

Miller Jr W. - One of the best experts on this subject based on the ideXlab platform.

  • Superintegrability and higher order constants for quantum systems
    'Pleiades Publishing Ltd', 2010
    Co-Authors: Kalnins E. G., Kress J. M., Miller Jr W.
    Abstract:

    We refine a method for finding a canonical form for symmetry operators of arbitrary order for the Schroedinger Eigenvalue Equation on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. As examples we treat two potentials with parameter k (one of which is the Tremblay, Turbiner, and Winternitz system) that have been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We also develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.Comment: 25 page

Kalnins E. G. - One of the best experts on this subject based on the ideXlab platform.

  • Superintegrability and higher order constants for quantum systems
    'Pleiades Publishing Ltd', 2010
    Co-Authors: Kalnins E. G., Kress J. M., Miller Jr W.
    Abstract:

    We refine a method for finding a canonical form for symmetry operators of arbitrary order for the Schroedinger Eigenvalue Equation on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. As examples we treat two potentials with parameter k (one of which is the Tremblay, Turbiner, and Winternitz system) that have been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We also develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.Comment: 25 page

Kress J. M. - One of the best experts on this subject based on the ideXlab platform.

  • Superintegrability and higher order constants for quantum systems
    'Pleiades Publishing Ltd', 2010
    Co-Authors: Kalnins E. G., Kress J. M., Miller Jr W.
    Abstract:

    We refine a method for finding a canonical form for symmetry operators of arbitrary order for the Schroedinger Eigenvalue Equation on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. As examples we treat two potentials with parameter k (one of which is the Tremblay, Turbiner, and Winternitz system) that have been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We also develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.Comment: 25 page

Valerio Magnasco - One of the best experts on this subject based on the ideXlab platform.

  • 3 – The Particle in the Box
    Elementary Methods of Molecular Quantum Mechanics, 2007
    Co-Authors: Valerio Magnasco
    Abstract:

    Publisher Summary The Schroedinger Equation can be solved exactly for few physical systems such as the free particle, the particle in the box or tunneling across a barrier, certain kinds of rotating bodies, the harmonic oscillator, the atomic 1-electron system, and the molecular 1-electron 2-centre problem. This chapter discusses the application of the principles of quantum mechanics to exactly solvable problems, by examining the particle in the box and the atomic 1-electron (or hydrogen-like) system. These two examples are exhaustive enough in explaining the general techniques of solution of the Schroedinger Eigenvalue Equation. The chapter also discusses the free particle in one and three dimensions, focusing on the 1-dimensional problem of a particle confined in a box. The 3-D box is a system whose potential energy is zero when the particle is within a closed region and is constant everywhere else.

  • 4 – The Hydrogen-Like System
    Elementary Methods of Molecular Quantum Mechanics, 2007
    Co-Authors: Valerio Magnasco
    Abstract:

    Publisher Summary This chapter discusses the hydrogen-like system. There are many reasons for choosing the hydrogen-like system as an example of exact solution of the Schroedinger Eigenvalue Equation. It is the only atomic case that can be solved exactly. The physically permissible solutions are the simplest example of atomic orbitals and their properties. Also, the mathematical techniques of solution of the relevant differential Equations are completely general and can be applied to the hydrogen atom in the electric or magnetic fields thus giving the exact evaluation of second-order properties such as the electric polarizabilities or the magnetic susceptibilities. The physical constraints imposed on the mathematical solutions explain the origin of quantum numbers, justifying the assumptions of the Bohr theory. Finally, the possibility of disposing of relatively simple solutions in exact analytical form is essential for checking unequivocally approximate solutions such as those obtained by use of the powerful variation theorem. The hydrogen-like system consists of a single electron of mass m2 and charge −e attracted at distance r by a nucleus of mass m1 and charge +Ze. It is an atomic 1-electron problem covering a series of isoelectronic physical systems depending on the value assumed by the nuclear charge Z. This chapter concludes by reviewing this problem in detail.