Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations

Fan, Hong-Yi; Lu, Hai-liang; Yang, Fan

doi:10.1016/j.aop.2005.09.011

Cited by 342 publications

(153 citation statements)

References 22 publications

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“…In this way the exact time-dependent wavefunction of the Schrödinger equation governed by the CK Hamiltonian can be directly obtained, which represents a squeezed number state. The corresponding Wigner function is derived by virtue of the Weyl ordered form of the Wigner operator [11] and the order-invariance of Weyl ordered operators under similar transformations [12].…”

Section: Introductionmentioning

confidence: 99%

Fresnel Operator, Squeezed State and Wigner Function for Caldirola–kanai Hamiltonian

2011

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Based on the technique of integration within an ordered product (IWOP) of operators we introduce the Fresnel operator for converting Caldirola-Kanai Hamiltonian into time-independent harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A, B, C, D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of partial differential equations set up in the above mentioned converting process. In this way the exact wavefunction solution of the Schrödinger equation governed by the Caldirola-Kanai Hamiltonian is obtained, which represents a squeezed number state. The corresponding Wigner function is derived by virtue of the Weyl ordered form of the Wigner operator and the order-invariance of Weyl ordered operators under similar transformations. The method used here can be suitable for solving Schrödinger equation of other time-dependent oscillators.

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Section: Introductionmentioning

confidence: 99%

Fresnel Operator, Squeezed State and Wigner Function for Caldirola–kanai Hamiltonian

2011

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“…The task can be carried out by implementation of techniques accomplishing ordering of non-commuting operators (see e.g. [20][21][22][23] and Refs therein). Solving operator equation then essentially becomes into a matter of combinatorics, and interesting findings in this respect may be revealed in fact, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-ordered basis under time-dependent canonical transformations

Gianfreda

Landolfi

2017

Theor Math Phys

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We address the problem of integrating operator equations concomitant with the dynamics of non autonomous quantum systems by taking advantage of the use of time-dependent canonical transformations. In particular, we proceed to a discussion in regard to basic examples of one-dimensional non-autonomous dynamical systems enjoying the property that their Hamiltonian can be mapped through a time-dependent linear canonical transformation into an autonomous form, up to a timedependent multiplicative factor. The operator equations we process essentially reproduce at the quantum level the classical integrability condition for these systems. Operator series form solutions in the Bender-Dunne basis of pseudo-differential operators for one dimensional quantum system are sought for such equations. The derivation of generating functions for the coefficients involved in the minimal representation of the series solutions to the operator equations under consideration is particularized. We also provide explicit form of operators that implement arbitrary linear transformations on the Bender-Dunne basis by expressing them in terms of the initial Weyl ordered basis elements. We then remark that the matching of the minimal solutions obtained independently in the two basis, i.e. the basis prior and subsequent the action of canonical linear transformation, is perfectly achieved by retaining only the lowest order contribution in the expression of the transformed Bender-Dunne basis elements.

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“…(11) and |00 00| = : exp(−a † 1 a 1 − a † 2 a 2 ) : and perfoming the integration in Eq. (10) by virtue of the IWOP technique [7,8], we obtain the normally ordered form of the Wigner operator ∆ (σ, γ)…”

Section: Wigner Function For Two-body Correlated Systemmentioning

confidence: 99%

“…Using the technique of integration within an ordered product of operators (IWOP) [7,8] and the vacuum projector |0 0| = : exp −a † a : ( : : denotes normal ordering), we have obtained the explicitly normal ordering form of ∆ (x, p) …”

Section: Introductionmentioning

confidence: 99%

Time evolution of Wigner functions governed by bipartite Hamiltonian system with kinetic coupling

Xu¹,

Liu²,

Yuan³

et al. 2010

Braz. J. Phys.

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Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations

Cited by 342 publications

References 22 publications

Fresnel Operator, Squeezed State and Wigner Function for Caldirola–kanai Hamiltonian

Fresnel Operator, Squeezed State and Wigner Function for Caldirola–kanai Hamiltonian

Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-ordered basis under time-dependent canonical transformations

Time evolution of Wigner functions governed by bipartite Hamiltonian system with kinetic coupling

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