Abstract
As a very important example for dynamical symmetries in the context ofq-generalized quantum mechanics the algebraaa†−q−2a†a=1is investigated. It represents the oscillator symmetrySUq(1,1)and is regarded as a commutation phenomenon of theq-Heisenberg algebra which provides a discrete spectrum of momentum and space,i.e., a discrete Hilbert space structure. Generalizedq-Hermite functions and systems of creation and annihilation operators are derived. The classical limitq→1is investigated. Finally theSUq(1,1)algebra is represented by the dynamical variables of theq-Heisenberg algebra.
Highlights
The dynamical symmetry SU(1, 1) plays an important role in the abstract description of quantum mechanical harmonic oscillators
In quantum mechanics the ground state of the harmonic oscillator is given by the equation (P i)[0) 0 or (p + i)[0) 2p[0) 0 (14)
As the described situation admits a direct approach to the a-at-relations in quantum mechanics we want to generalize it in the qdeformed case
Summary
The dynamical symmetry SU(1, 1) plays an important role in the abstract description of quantum mechanical harmonic oscillators. Following the approach of Wess it has turned out that this Hilbert space is directly related to a quantization of momentum and space via the parameter q, see [10]. Questions concerning the limit were investigated in detail in [12] where is shown that a suitable choice for the q-deformed Heisenberg algebra is p qp -iq3/Zu (10). To deal with the mentioned non trivial situation in the q-case we will start in the second chapter from an observation which we call commutator symmetry or oscillator symmetry of Heisenberg algebras. Making use of it we can generalize the oscillator concept of quantum mechanics. Further investigations concerning the deeper physical meaning of this situation are presently prepared [3]
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