q-Deformed dynamics and Virial theorem

Zhang, Jian-zu

doi:10.1016/s0370-2693(02)02064-6

Cited by 3 publications

(1 citation statement)

References 32 publications

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“…Then we can find the fact that the simplest case of the above formula in case of k = 2 and m = n is the Fay identity of the discrete KP hierarchies, so we will give the proof that the discrete KP hierarchies are equivalent to the addition formula. The q-deformed KP hierarchy is also an important hot topic in the research of the integrable system, which is based on quantum calculus [12,13] and related to the q-deformed quantum integrable models [14], the q-deformed Bose gas [15], the q-deformed thermodynamics and Virial theorem [16,17], etc. The q-KP hierarchy can be viewed as differential equations for the tau function τ q (x; t).…”

Section: Introductionmentioning

confidence: 99%

Addition Formulae of Discrete KP, q-KP and Two-Component BKP Systems

Gao¹,

Li²,

He³

2016

Commun. Theor. Phys.

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Abstract. In this paper, we constructed the addition formulae for several integrable hierarchies, including the discrete KP, the q-deformed KP, the two-component BKP and the D type Drinfeld-Sokolov hierarchies. With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies, we prove that these addition formulae are equivalent to these hierarchies. These studies show that the addition formula in the research of the integrable systems has good universality.

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Section: Introductionmentioning

confidence: 99%

Addition Formulae of Discrete KP, q-KP and Two-Component BKP Systems

Gao¹,

Li²,

He³

2016

Commun. Theor. Phys.

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Structures of q-deformed currents

Zhang

2003

Physics Letters B

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The non-perturbation and perturbation structures of the q-deformed probability currents are studied. According to two ways of realizing the q-deformed Heisenberg algebra by the undeformed operators, the perturbation structures of two q-deformed probability currents are explored in detail. Locally the structures of two perturbation q-deformed probability currents are different, one is explicitly potential dependent; the other is not. But their total contributions to the whole space are the same.Recently the q-deformed quantum theory, as a possible modification of the ordinary quantum theory at extremely small space scale, say, much smaller than 10 −18 cm, has obtained attention. In literature different frameworks of q-deformed quantum theories were established . In order to establish a consistent framework in q-deformed quantum theories three delicate points must be considered: the construction of the simultaneous hermitian position and momentum operators; the establishment of the correspondence of the position and momentum operators to the q-deformed annihilation and creation operators; and the reduction of the q-deformed annihilation and creation operators to the undeformed ones. In the framework of the q-deformed Heisenberg algebra developed in Refs. [2, 4] the above three aspects are investigated in detail. This framework is self-consistent. New features, both in the uncertainty relations and dynamics, in this framework are explored. The q-deformed uncertainty relation essentially deviates from the Heisenberg one [14, 15, 17, 21]: Heisenberg's minimal uncertainty relation is undercut. In a special q-deformed squeezed states a new critical phenomenon is explored[17]: at a critical point the variance of one component of a quadrature of light field approaches zero, but the variance of the conjugate component remains finite. Such critical phenomenon is forbidden by Heisenberg's uncertainty relations, but allowed by the q-deformed uncertainty relations. In dynamics the non-perturbation energy spectrum of the q-deformed Schrödinger equation exhibits an exponential structure [3, 4, 16] with new degrees of freedom and new quantum numbers.Using such an exponential structure the spectrum of quark-lepton is explained [16]. In the perturbation aspects the q-deformed dynamics also exhibits a new feature: the perturbation expansion of the q-deformed Hamiltonian possesses complicated structures, which amount to additional momentum-dependent interactions [2,4,14,16,18,19,21]. Furthermore, corresponding to two ways of realizing the q-deformed operators by the undeformed ones there are two q-perturbation momentum-dependent Hamiltonians, one originates from the perturbation expansion of the potential in one configuration space, the other originates from the perturbation expansion of the kinetic energy in the other configuration space.At the level of operators, they are different. But they contribute the same shifts to the undeformed energy spectrum. [18,19,21].In this paper the non-perturbation and perturbation structures of the...

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On theq-deformed coherent states of a generalizedf-oscillator

Marchiolli

2005

Phys. Scr.

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Adopting the framework of the generalized q-deformed Heisenberg-Weyl algebra U (α,β,γ ) q (h 4 ), we present a mathematical procedure which leads us to obtain analytical expressions for a general class of q-deformed coherent states associated with the different patterns of the energy spectrum exhibited by the nonlinear f-oscillator. In particular, we establish the properties of a small group of q-deformed coherent states for α > γ > 0 with emphasis on the resolution of unity. As an application of these properties, we investigate the Robertson-Schrödinger uncertainty relation and the squeezing effect for the deformed coordinate and momentum operators, which are defined in terms of the abstract elements of this algebra. Furthermore, we also obtain the Wigner function and the correct quantum-mechanical marginal distributions in phase space.

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q-Deformed dynamics and Virial theorem

Cited by 3 publications

References 32 publications

Addition Formulae of Discrete KP, q-KP and Two-Component BKP Systems

Addition Formulae of Discrete KP, q-KP and Two-Component BKP Systems

Structures of q-deformed currents

On theq-deformed coherent states of a generalizedf-oscillator

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