Showing papers on "Coherent states in mathematical physics published in 1992"
TL;DR: In this article, minimum uncertainty coherent states and annihilation operator coherent states for the Morse oscillator were derived and shown to be equivalent in the limit of small anharmonicity constant.
Abstract: Minimum uncertainty coherent states and annihilation operator coherent states for the Morse oscillator are derived and shown to be equivalent. They reduce, in the limit of small anharmonicity constant, or, equivalently, in the limit of large well depth, to the approximate coherent states derived previously from the use of generalized displacement operator.
55 citations
TL;DR: In this article, the algebra of q-fermion operators is re-examined and generalized q-oscillators defined for - infinity (q
Abstract: The algebra of q-fermion operators, developed earlier by two of the present authors is re-examined. It is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q=1. It is shown that it is possible to introduce generalized q-oscillators defined for - infinity (q
53 citations
TL;DR: In this paper, the Hermitian conjugation of a q-differential operator is discussed on a one-dimensional lattice without uniform interval, and a deformation of quantum mechanics in one dimension is presented.
Abstract: On a one-dimensional lattice without uniform interval, the Hermitian conjugation of a q-differential operator is discussed. Then a deformation of quantum mechanics in one dimension is presented. As an application, the harmonic oscillator is discussed. The energy spectrum and the eigenfunctions are shown to depend on an arbitrary deformation function. The deformed coherent states are also discussed. It is found that the completeness relation of coherent states holds for the case of q-coherent states, i.e. the deformation of the Heisenberg-Weyl algebra is a q-analogue Hopf algebra.
19 citations
TL;DR: The application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored.
Abstract: Fock states and coherent states have been widely applied in analyses of quantum-optics experiments In this paper, the application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency This choice of basis functions will be used to analyze the interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic-field mode Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the \ensuremath{\Delta}m=0 selection rule The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system The diamagnetic term, ${\mathit{e}}^{2}$${\mathbf{A}}^{2}$/2${\mathit{mc}}^{2}$, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequencyAn approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency This expression indicates that inclusion of the RWA terms adds a small ``ripple'' on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant The ripple is interpreted as ``interference'' between two displaced oscillator-field states that represent the unperturbed state When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients
18 citations
TL;DR: In this article, the authors apply the coherent states of the Lie group H4⊗SU(2) to define quantum surfaces of section for a model involving centrifugal coupling of a harmonic bend with molecular rotation, and study two harmonic normal modes coupled to overall molecular rotation through coriolis interaction.
Abstract: We describe our work on qualitative methods for visualizing the quantum eigenstates of systems with nonlinear classical dynamics. For two-degree-of-freedom systems, our approach is based on the use of generalized coherent states, and allows systems with nonoscillator kinematics to be investigated. The general approach is illustrated with two examples involving vibration-rotation interaction in polyatomic molecules. We apply the coherent states of the Lie groupH
4⊗SU(2) to define quantum surfaces of section for a model involving centrifugal coupling of a harmonic bend with molecular rotation, andSU(2)⊗SU(2) coherent states to study two harmonic normal modes coupled to overall molecular rotation through coriolis interaction. In both systems, quantum states are visualized on the rotational surface of section and compared with the corresponding classical phase space structure. Striking classical-quantum correspondence is observed. We then describe recent results on the quantum states of (N⩾ 3)-dimensional systems of coupled nonlinear oscillators, which reveal a quantum delocalization that is reminiscent of classical Arnold diffusion.
17 citations
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TL;DR: In this article, the Bargmann Fock construction can also be done in the quantum group symmetric case, which leads to a 'q-deformed quantum mechanics' in which the basic concepts, Hilbert space of states and unitarity of time evolution, are preserved.
Abstract: Usually in quantum mechanics the Heisenberg algebra is generated by operators of position and momentum. The algebra is then represented on an Hilbert space of square integrable functions. Alternatively one generates the Heisenberg algebra by the raising and lowering operators. It is then natural to represent it on the Bargmann Fock space of holomorphic functions. In the following I show that the Bargmann Fock construction can also be done in the quantum group symmetric case. This leads to a 'q- deformed quantum mechanics' in which the basic concepts, Hilbert space of states and unitarity of time evolution, are preserved.
4 citations
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TL;DR: In this article, the authors present a program to obtain and understand coherent states for general systems, including supersymmetric systems, by analyzing the role of Grassmann numbers in quantum mechanics.
Abstract: This article reports on a program to obtain and understand coherent states for general systems. Most recently this has included supersymmetric systems. A byproduct of this work has been studies of squeezed and supersqueezed states. To obtain a physical understanding of these systems has always been a primary goal. In particular, in the work on supersymmetry an attempt to understand the role of Grassmann numbers in quantum mechanics has been initiated.
4 citations
TL;DR: An explicit expression for the coherent states is obtained and the mean values of the energy and the field with its standard deviations are calculated; the classical values are recovered in the limit of a small fine structure constant.
Abstract: In this work we present a study of the one-photon wave function and the coherent states in the formulation of second quantization of free electromagnetism in loop space. We obtain an explicit expression for the coherent states and calculate the mean values of the energy and the field with its standard deviations; we recover the classical values in the limit of a small fine structure constant.
2 citations
TL;DR: In this article, the functional representation of the Hilbert space in terms of coherent states is reconsidered with the purpose of studying the connection between quantum states and the corresponding distributions of classical statistical mechanics.
Abstract: The functional representation of the Hilbert space in terms of coherent states is reconsidered with the purpose of studying the connection between quantum states and the corresponding distributions of classical statistical mechanics. The statistical predictions of both theories are compared and the relevance of these results for the conventional interpretation of quantum mechanics is discussed.
2 citations
01 Feb 1992
TL;DR: In this paper, a general definition of squeezed states is proposed and its main features are illustrated through a discussion of the standard optical coherent states represented by Gaussian pure states, and the set-up involves representations of groups on Hilbert spaces over homogeneous spaces of the group, and relies on the construction of a square integrable (coherent state) group representation modulo a subgroup.
Abstract: A general definition of squeezed states is proposed and its main features are illustrated through a discussion of the standard optical coherent states represented by 'Gaussian pure states'. The set-up involves representations of groups on Hilbert spaces over homogeneous spaces of the group, and relies on the construction of a square integrable (coherent state) group representation modulo a subgroup. This construction depends upon a choice of a Borel section which has a certain permissible arbitrariness in its selection; this freedom is attributable to a squeezing of the defining coherent states of the representation, and corresponds in this way to a sort of gauging.
14 Aug 1992
TL;DR: The phase-space-picture approach to quantum non-equilibrium statistical mechanics via the characteristic function of infinite-mode squeezed coherent states is introduced in this article, which provides an interesting geometrical interpretation of quantum nonequilibrium phenomena.
Abstract: The phase-space-picture approach to quantum non-equilibrium statistical mechanics via the characteristic function of infinite-mode squeezed coherent states is introduced. We use quantum Brownian motion as an example to show how this approach provides an interesting geometrical interpretation of quantum non-equilibrium phenomena.