This work describes coherent states for a physical system governed by a Hamiltonian operator, in two dimensional space, of spinless charged particles subject to a perpendicular magnetic field B, coupled with a harmonic potential. The underlying su(1, 1) Lie algebra and Barut-Girardello coherent states are constructed and discussed. Then, the Berezin -Klauder -Toeplitz quantization, also known as coherent state (or anti-Wick) quantization, is discussed. The thermodynamics of such a quantum gas system is elaborated and analyzed.
Quantum versions of cylindric phase space, like for the motion of a particle on the circle, are obtained through different families of coherent states. The latter are built from various probability distributions of the action variable. The method is illustrated with Gaussian distributions and uniform distributions on intervals, and resulting quantizations are explored.
The behavior of an electron in an external uniform electromagnetic background coupled to an harmonic potential, with noncommuting space coordinates, is considered in this work. The thermodynamics of the system is studied. Matrix vector coherent states (MVCS) as well as quaternionic vector coherent states (QVCS), satisfying required properties, are also constructed and discussed.
Since the introduction of the concept of canonical coherent states (CS), associated with the one dimensional harmonic oscillator by Schrödinger in 1926 1 , followed decades in which this concept has reached great investigations 2 -5 . CS are a useful mathematical framework for dealing with the connection between classical and quantum mechanics 5-7 . These states can globally be constructed in three equivalent ways: (i) by defining them as eigenstates of the lowering operator (called CS of the Barut-Girardello 8 type), (ii) by applying a unitary displacement operator on a ground state (Klauder-Perelomov CS 9 or CS of the Gazeau-Klauder type), and (iii) by considering them as quantum states with a minimum uncertainty relationship 10,11 .The CS for shape-invariant potentials (SIP) performed in this work 12,13 , belong to the Barut-Girardello type. They are built using algebraic approach based on the supersymmetric quantum mechanics (SUSY-QM) 14,15 . SUSY QM deals with the study of partner Hamiltonians which are isospectral, that is, they have almost the same energy eigenvalues. A number of such partner Hamiltonians satisfy an integrability condition called shape invariance 16 -18 .However, not all exactly solvable systems are shape-invariant.Recently, a considerable attention is devoted to photon-added CS (PACS) 19 -22 , first introduced by Agarwal and Tara 23 . The PACS represent interesting states generalizing both the Fock states and CS. Indeed, they are obtained by repeatedly operating the photon creation operator on an ordinary CS. In some previous works, the PACS were assimilated to nonlinear CS. Their various generalizations were also performed 24,25 . They evidence some nonclassical effects, for e.g, amplitude squeezing, sub-Poissonian behaviour, nonclassical quasi-probability distribution. In one of our previous papers 26 , a family of photon added as well as photon depleted CS related to the inverse of ladder operators acting on hypergeometric CS was introduced. Their squeezing and antibunching properties were investigated in both standard (nondeformed) and deformed quantum optics. Recently 27 , new generalized PACS were formulated by excitations on a family of generalized CS. Their non-classical features and quantum statistical properties were compared with those obtained by Agarwal. Besides, in another paper 28 , photon-subtracted generalized CS, which are reminiscent of the PACS, were introduced; their nonclassical features were also discussed. PACS find many applications in physics. In 29 , a generating Schrödinger-cat-like states of a single-mode
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