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arXiv:0712.2078

Algebraic Solution of the Harmonic Oscillator With Minimal Length Uncertainty Relations

Kay L. Gemba, Zvonimir Hlousek, Zoltán Papp

Published Dec 13, 2007• 2 citations
arXivHTMLPDF

Abstract

In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrödinger equation of the model. The approach can be generalized to the $D$-dimensional oscillator with non-commuting coordinates.

DOI
10.48550/arxiv.0712.2078

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In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrödinger equation of the model. The approach can be generalized to the DDD-dimensional oscillator with non-commuting coordinates.
We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by $[\stackrel{^}{x},\stackrel{^}{p}]=i\ensuremath{\hbar}(1+\ensuremath{\beta}{\stackrel{^}{p}}^{2})$. This deformed commutation relation leads to the minimal length uncertainty relation $\ensuremath{\Delta}x\ensuremath{\ge}(\ensuremath{\hbar}/2)(1/\ensuremath{\Delta}p+\ensuremath{\beta}\ensuremath{\Delta}p)$, which implies that $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ at small $\ensuremath{\Delta}p$ while $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$ at large $\ensuremath{\Delta}p$. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0m>0m>0), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 (2002), only populate the $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ branch. The other branch, $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$, is found to be populated by the energy eigenstates of the ``inverted'' harmonic oscillator (m<0m<0m<0). The Hilbert space in the inverted case admits an infinite ladder of positive energy eigenstates provided that $\ensuremath{\Delta}{x}_{\mathrm{min}}=\ensuremath{\hbar}\sqrt{\ensuremath{\beta}}>\sqrt{2}[{\ensuremath{\hbar}}^{2}/k|m|{]}^{1/4}$. Correspondence with the classical limit is also discussed.
Several quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose their physical meaning. In quantum mechanics, the insurgence of such a minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on position measurements has a lower bound. The value of the minimal length is not predicted by theories and must be estimated experimentally. In this paper, we address the quantum bound to the estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra induced by the deformed commutation relations.
We construct the deformed ladder operators in the presence of a minimal length to study the one- and two-mode squeezed harmonic oscillator. The generalized Hamiltonian of the system is expressed in terms of a deformed su(1,1)su(1,1)su(1,1) algebra. The realizations of this algebra allow us to convert the purely quantum mechanical problem of the model into a differential equation. By means of the Nikiforov–Uvarov method, the energy eigenvalues are obtained and the corresponding wave functions, in the momentum space, are expressed in terms of hypergeometric functions. Our study shows that the domain of existence of the energy levels is extended and this extension is due to the deformation parameter.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
In the weak deformation (WD) approximation of the Weyl–Heisenberg algebra, the corresponding generalized coherent states and displacement operator are constructed. It is shown that those states, and contrary to the non-deformed Weyl-Heisenberg algebra, are not eigenstates of the annihilation operator. Moreover, and as an alternative to the Chaïchian et al. Q-deformed path integral approach (where Q is the deformation parameter), using the Bargmann Fock representation, we propose in the WD approximation, a general simple formalism. As an application, we calculate the propagator and the wave function of the harmonic oscillator.PACS Nos.: 03.65.Fd, 31.15.Kb