Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom

Lorente, Miguel

doi:10.1016/s0375-9601(01)00303-6

Cited by 29 publications

(18 citation statements)

References 8 publications

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“…At this stage, new families of discrete Clifford‐valued polynomials, for example, hypercomplex generalizations of Kravchuk polynomials (cf. ) should appear.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

(Discrete) Almansi type decompositions: an umbral calculus framework based on osp(1|2) symmetries

Faustino

Ren

2011

Math. Meth. Appl. Sci.

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Communicated by Svetlin GeorgievWe introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials IROEx shall be described in terms of the generators of the Weyl-Heisenberg algebra. The extension of IROEx to the algebra of Clifford-valued polynomials P gives rise to an algebra of Clifford-valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra osp.1j2/.This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of

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“…At this stage, new families of discrete Clifford‐valued polynomials, for example, hypercomplex generalizations of Kravchuk polynomials (cf. ) should appear.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

(Discrete) Almansi type decompositions: an umbral calculus framework based on osp(1|2) symmetries

Faustino

Ren

2011

Math. Meth. Appl. Sci.

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“…This last model, too, recovers the results for the ordinary harmonic oscillator in an appropriate limit. As do the discrete models of the quantum harmonic oscillator in terms of Kravchuk polynomials [10] or Harper functions [4]. Summarizing, the correct continuous limit of discrete models can be obtained in many situations.…”

Section: Discussionmentioning

confidence: 99%

Properties of finite Gaussians and the discrete-continuous transition

Cotfas¹,

Dragoman²

2012

J. Phys. A: Math. Theor.

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Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finitedimensional systems. Based on Weyl's approach, generalized by Schwinger, a selfconsistent theoretical framework describing physical systems characterised by a finitedimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension d of the system increases.

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“…Therefore, the operators A ± , A, e B AE , e B, e C AE ; e D AE close an algebra given by the su(2) and su(1, 1) subalgebras (22), (59), (80), (86) and the non-vanishing crossing commutators,…”

Section: The Trigonometric Pt Dynamical Algebramentioning

confidence: 99%

“…So, in order to circumvent this problem, a second way is to identify the eigenfunctions, including the normalization constants, in terms of some known special functions. Then, the ladder operators can be found by means of certain recurrence relations of these functions [7,[18][19][20][21][22][23]. For example, the ''dynamical potential algebras" of Pöschl-Teller (PT) potentials have been found by connecting its eigenfunctions with Wigner rotation matrices [24].…”

Section: Introductionmentioning

confidence: 99%