The Wigner distribution function for the su(2) finite oscillator and Dyck paths
 Author
 Roy Oste (UGent) and Joris Van der Jeugt (UGent)
 Organization
 Abstract
 Recently, a new definition for a Wigner distribution function for a onedimensional finite quantum system, in which the position and momentum operators have a finite (multiplicityfree) spectrum, was developed. This distribution function is defined on discrete phasespace (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the preWigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the su(2) finite oscillator. The first expression for the matrix elements involves sums over squares of Krawtchouk polynomials, and follows from standard techniques. We also manage to present a second solution, where the matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are defined in terms of the wellknown Dyck paths. This combinatorial expression of the preWigner matrix elements turns out to be particularly simple.
 Keywords
 Dyck path, finite oscillator, Wigner distribution function
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU5671017
 MLA
 Oste, Roy, and Joris Van der Jeugt. “The Wigner Distribution Function for the Su(2) Finite Oscillator and Dyck Paths.” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, vol. 47, no. 28, 2014, doi:10.1088/17518113/47/28/285301.
 APA
 Oste, R., & Van der Jeugt, J. (2014). The Wigner distribution function for the su(2) finite oscillator and Dyck paths. JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, 47(28). https://doi.org/10.1088/17518113/47/28/285301
 Chicago authordate
 Oste, Roy, and Joris Van der Jeugt. 2014. “The Wigner Distribution Function for the Su(2) Finite Oscillator and Dyck Paths.” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL 47 (28). https://doi.org/10.1088/17518113/47/28/285301.
 Chicago authordate (all authors)
 Oste, Roy, and Joris Van der Jeugt. 2014. “The Wigner Distribution Function for the Su(2) Finite Oscillator and Dyck Paths.” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL 47 (28). doi:10.1088/17518113/47/28/285301.
 Vancouver
 1.Oste R, Van der Jeugt J. The Wigner distribution function for the su(2) finite oscillator and Dyck paths. JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL. 2014;47(28).
 IEEE
 [1]R. Oste and J. Van der Jeugt, “The Wigner distribution function for the su(2) finite oscillator and Dyck paths,” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, vol. 47, no. 28, 2014.
@article{5671017, abstract = {{Recently, a new definition for a Wigner distribution function for a onedimensional finite quantum system, in which the position and momentum operators have a finite (multiplicityfree) spectrum, was developed. This distribution function is defined on discrete phasespace (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the preWigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the su(2) finite oscillator. The first expression for the matrix elements involves sums over squares of Krawtchouk polynomials, and follows from standard techniques. We also manage to present a second solution, where the matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are defined in terms of the wellknown Dyck paths. This combinatorial expression of the preWigner matrix elements turns out to be particularly simple.}}, articleno = {{285301}}, author = {{Oste, Roy and Van der Jeugt, Joris}}, issn = {{17518113}}, journal = {{JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL}}, keywords = {{Dyck path,finite oscillator,Wigner distribution function}}, language = {{eng}}, number = {{28}}, pages = {{16}}, title = {{The Wigner distribution function for the su(2) finite oscillator and Dyck paths}}, url = {{http://dx.doi.org/10.1088/17518113/47/28/285301}}, volume = {{47}}, year = {{2014}}, }
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