One-dimensional harmonic oscillator problem and its hidden \(SU(1,1)\) symmetry in the presence of a minimal length. (English) Zbl 1448.81371
Summary: The one-dimensional harmonic oscillator in the presence of a minimal length possesses a hidden \(SU(1,1)\) symmetry. This symmetry makes it possible to construct the deformed canonical coherent states. These states turn out to be ground states of the deformed dipolar coupling. Indeed, the coupling to the external field may be considered as a dipolar coupling of a charged dipole to an electric field in one dimension. The spectrum is constructed algebraically. The wave functions in the momentum space are then reported. We have found that the momentum probability density of order \(n\) possesses \(n+1\) vertices and these vertices of excited states disappear from \(n-1\) when the deformation parameter increases. So, the particle in these states loses its point nature and can even explore space. The Heisenberg uncertainty relation involving a non-zero minimal length induces then a finite lower bound to spatial localisation.
MSC:
| 81R30 | Coherent states |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
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