Quantum algebras and parity-dependent spectra. (English) Zbl 1132.81348
Summary: We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.
MSC:
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 46L60 | Applications of selfadjoint operator algebras to physics |
Online Encyclopedia of Integer Sequences:
Tangent (or ”Zag”) numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).Counts a family of permutations occurring in the study of squeezed states of the simple harmonic oscillator.
Recursive triangle for calculating A186491.
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