|M.Sc Student||Gur Yaniv|
|Subject||SO(2,1) Lie Algebra and Radial Coherent States - from the|
Harmonic Oscillator to the Hydrogen Atom
|Department||Department of Physics||Supervisor||Professor Ady Mann|
Coherent states are widely used in many areas of quantum physics, e.g., quantum optics, quantum information and the study of the quantum-classical correspondence. In this thesis we mainly study coherent states as a tool to discuss quantum-classical correspondence. We focus on radial coherent states where we
construct Barut-Girardello and Perelomov coherent states (BGCS and PCS, respectively) for the isotropic harmonic oscillator (IHO) in radial coordinates and in arbitrary dimension. The construction of these states is associated with the algebra of the non-compact group SO(2,1). These states are expanded as a superposition of all radial eigenfunctions of the IHO with constant l (angular momentum). Each of these superpositions may be written in closed form using the Laguerre generating functions, where each of the sets corresponds to a different generating function and results in a different set of states. We discuss the
time evolution properties of these states with respect to their classical properties and with comparison to the usual coherent states of the one-dimensional harmonic oscillator.
Through the algebra of SO(2,1) we constructed the relation between the IHO and the Coulomb problem in any dimension.We take advantage of the mapping between the IHO and the Coulomb problem to transform directly the radial BGCS of the IHO into the radial BGCS of the Sturm-Coulomb problem, namely, the generating function of the Sturm functions. Then we define "physical" BGCS by acting on the Sturm-Coulomb BGCS with the dilatation operator and then fix their normalization factor. We explore the dynamical properties of these states and show that they evolve periodically with respect to a fictitious time without dispersing.