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Cited by 9Quantum Mechanics and ApplicationsQuantum chaos and dynamical systemsSpectral Theory in Mathematical Physics

Linear canonical transformations and quantum phase: a unified canonical and algebraic approach

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Hakioglu, T·1999·Journal of Physics A: Mathematical and General
DOI 10.1088/0305-4470/32/22/312
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Abstract

The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwinger's unitary-canonical operator basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with dynamical symmetry is examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.

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  • Assuming ω is rational, the energy eigenstates of the Bloch electron problem are given by ψ(X,Y) = e^{2πikY/n} φ(X), where φ solves a difference equation.

    “Assuming that ω is rational, let us write write ω = m/n where m is an integer, and n is a positive integer coprime to m.The energy eigenstates ψ with energy eigenvalue E are given by ψ(X, Y ) = e 2π ikY/n φ(X) where k is an integer, and φ is a solution to φ(X -1) + φ(X + 1) + 2 cos(2π(mX + k)/n)φ(X) = Eφ(X).”

    From The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform (2000) · cites this paper as [16]

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Research works printed but not linked: 10

  1. Dirac P A M 1927 Proc. R. Soc. A 114 243
  2. Dirac P A M 1958 Principles of Quantum Mechanics (Oxford: Oxford University Press)
  3. Mandel L and Wolf E 1995 Optical Coherence and Quantum Optics (Cambridge: Cambridge University Press)
  4. Schwinger J 1960 Proc. Natl Acad. Sci. 46 883
  5. Schwinger J 1960 Proc. Natl Acad. Sci. 46 1401
  6. Arnold V 1978 Mathematical Methods of Classical Mechanics (Berlin: Springer)
  7. Ellinas D and Floratos E G 1998 Prime decomposition and entanglement measure of finite quantum systems Preprint quant-ph/9806007 · arXiv
  8. Candan Ç, Alper Kutay M and Ozaktas H M 1999 The discrete fractional Fourier transformation Proc. IEEE Int. Conf. Acoustic, Speech and Signal Processing (Piscataway, NJ: IEEE) pp 1713-6
  9. Gilmore R 1974 Lie Groups, Lie Algebras and Some of Their Applications (New York: Wiley)
  10. Moyal J 1949 Proc. Camb. Phil. Soc. 45 99

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Authors
Hakioglu, T.
Affiliations
Bilkent University (TR)
Corresponding author
Hakioglu, T
Venue
Journal of Physics A: Mathematical and General
Publisher
IOP Publishing
Published
1999-01-01
Volume
32
Issue
22
Pages
4111-4130
ISSN
  • 0305-4470 (ISSN-L)
  • 1361-6447
DOI
10.1088/0305-4470/32/22/312
OpenAlex
W2038153783
Cited by
9
References
43
Citations by year
2017: 2
FWCI
1.12
Type
journal article
Language
en
Open access
Locations
  • Journal of Physics A Mathematical and General (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Bilkent University Institutional Repository (Bilkent University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Orta Doğu Teknik Üniversitesi
  • Bilkent Üniversitesi
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
  • Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
Keywords
linear canonical transformations, quantum phase, Wigner function, quantum harmonic oscillator, dynamical symmetry, Hamiltonian systems, quantum phase space

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