Analytically Solvable Quantum Hamiltonians and Relations to Orthogonal Polynomials
- Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent (Belgium)
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.
- OSTI ID:
- 21366968
- Journal Information:
- AIP Conference Proceedings, Vol. 1243, Issue 1; Conference: 8. international workshop on Lie theory and its applications in physics, Varna (Bulgaria), 15-21 Jun 2009; Other Information: DOI: 10.1063/1.3460184; (c) 2010 American Institute of Physics; ISSN 0094-243X
- Country of Publication:
- United States
- Language:
- English
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GENERAL PHYSICS
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DATA TRANSMISSION
EIGENVALUES
EIGENVECTORS
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HARMONIC OSCILLATORS
LIE GROUPS
MATHEMATICAL SOLUTIONS
MATRICES
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QUANTIZATION
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SPIN
TRANSITION AMPLITUDES
AMPLITUDES
ANGULAR MOMENTUM
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