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Analytically solvable quantum Hamiltonians and relations to orthogonal polynomials

Gilles Regniers UGent and Joris Van der Jeugt UGent (2010) AIP CONFERENCE PROCEEDINGS. 1243. p.99-114
abstract
Quantum systems consisting of a linear chain of n harmonic oscillators coupled by a quadratic nearest-neighbour interaction are considered. We investigate when such a system is analytically solvable, in the sense that the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This leads to a relation with Jacobi matrices of systems of discrete orthogonal polynomials. Our study is first performed in the case of canonical quantization. Then we consider these systems under Wigner quantization, leading to solutions in terms of representations of Lie superalgebras. Finally, we show how such analytically solvable Hamiltonians also play a role in another application, that of spin chains used as communication channels in quantum computing. In this context, the analytic solvability leads to closed form expressions for certain transition amplitudes.
Please use this url to cite or link to this publication:
author
organization
year
type
conference
publication status
published
subject
keyword
quantum computation, Wigner quantization, quantum system, canonical quantization, transition amplitude, discrete orthogonal polynomial, Lie superalgebra, Jacobi matrices, analytically solvable quantum Hamiltonian, spin chains, harmonic oscillator
in
AIP CONFERENCE PROCEEDINGS
AIP Conf. Proc.
editor
Vladimir Dobrev
volume
1243
issue title
Lie theory and its applications in physics
pages
99 - 114
publisher
American Institute of Physics (AIP)
place of publication
Melville, NY, USA
conference name
8th International workshop on Lie Theory and its Applications in Physics
conference location
Varna, Bulgaria
conference start
2009-06-15
conference end
2009-06-21
Web of Science type
Proceedings Paper
Web of Science id
000282673700009
ISSN
0094-243X
ISBN
9780735407886
DOI
10.1063/1.3460184
language
English
UGent publication?
yes
classification
P1
copyright statement
I have transferred the copyright for this publication to the publisher
id
1044750
handle
http://hdl.handle.net/1854/LU-1044750
date created
2010-09-22 14:39:50
date last changed
2010-11-17 11:05:19
@inproceedings{1044750,
  abstract     = {Quantum systems consisting of a linear chain of n harmonic oscillators coupled by a quadratic nearest-neighbour interaction are considered. We investigate when such a system is analytically solvable, in the sense that the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This leads to a relation with Jacobi matrices of systems of discrete orthogonal polynomials. Our study is first performed in the case of canonical quantization. Then we consider these systems under Wigner quantization, leading to solutions in terms of representations of Lie superalgebras. Finally, we show how such analytically solvable Hamiltonians also play a role in another application, that of spin chains used as communication channels in quantum computing. In this context, the analytic solvability leads to closed form expressions for certain transition amplitudes.},
  author       = {Regniers, Gilles and Van der Jeugt, Joris},
  booktitle    = {AIP CONFERENCE PROCEEDINGS},
  editor       = {Dobrev, Vladimir},
  isbn         = {9780735407886},
  issn         = {0094-243X},
  keyword      = {quantum computation,Wigner quantization,quantum system,canonical quantization,transition amplitude,discrete orthogonal polynomial,Lie superalgebra,Jacobi matrices,analytically solvable quantum Hamiltonian,spin chains,harmonic oscillator},
  language     = {eng},
  location     = {Varna, Bulgaria},
  pages        = {99--114},
  publisher    = {American Institute of Physics (AIP)},
  title        = {Analytically solvable quantum Hamiltonians and relations to orthogonal polynomials},
  url          = {http://dx.doi.org/10.1063/1.3460184},
  volume       = {1243},
  year         = {2010},
}

Chicago
Regniers, Gilles, and Joris Van der Jeugt. 2010. “Analytically Solvable Quantum Hamiltonians and Relations to Orthogonal Polynomials.” In Aip Conference Proceedings, ed. Vladimir Dobrev, 1243:99–114. Melville, NY, USA: American Institute of Physics (AIP).
APA
Regniers, G., & Van der Jeugt, J. (2010). Analytically solvable quantum Hamiltonians and relations to orthogonal polynomials. In V. Dobrev (Ed.), AIP CONFERENCE PROCEEDINGS (Vol. 1243, pp. 99–114). Presented at the 8th International workshop on Lie Theory and its Applications in Physics, Melville, NY, USA: American Institute of Physics (AIP).
Vancouver
1.
Regniers G, Van der Jeugt J. Analytically solvable quantum Hamiltonians and relations to orthogonal polynomials. In: Dobrev V, editor. AIP CONFERENCE PROCEEDINGS. Melville, NY, USA: American Institute of Physics (AIP); 2010. p. 99–114.
MLA
Regniers, Gilles, and Joris Van der Jeugt. “Analytically Solvable Quantum Hamiltonians and Relations to Orthogonal Polynomials.” Aip Conference Proceedings. Ed. Vladimir Dobrev. Vol. 1243. Melville, NY, USA: American Institute of Physics (AIP), 2010. 99–114. Print.