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arXiv:2604.27018

Ground-state energy of a particle in a space with minimal length and minimal momentum

Arsen Panas, Volodymyr Tkachuk

quant-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• Revised May 4, 2026 (v2)
v1 · Apr 29, 2026v2 · May 4, 2026
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Abstract

In this article, we derived a rigorous lower bound on the ground-state energy for a class of one-dimensional quantum systems in deformed space with minimal coordinate and momentum uncertainties, representing the absolute minimum energy that is physically attainable. We considered a harmonic oscillator in such a space and calculate its ground-state energy. We generalized the problem to an a sufficiently broad class of potentials, deriving an equation for the coordinate uncertainty corresponding to the minimal energy, which can be solved numerically. Using a linear approximation in the deformation parameters, we obtained a general expression for the ground-state energy. We determined the domain of existence of solutions for the anharmonic oscillator potential with respect to the deformation parameters.

Comments
23 pages, 4 figures

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In this article, we derived a rigorous lower bound on the ground-state energy for a class of one-dimensional quantum systems in deformed space with minimal coordinate and momentum uncertainties, representing the absolute minimum energy that is physically attainable. We considered a harmonic oscillator in such a space and calculate its ground-state energy. We generalized the problem to an a sufficiently broad class of potentials, deriving an equation for the coordinate uncertainty corresponding to the minimal energy, which can be solved numerically. Using a linear approximation in the deformation parameters, we obtained a general expression for the ground-state energy. We determined the domain of existence of solutions for the anharmonic oscillator potential with respect to the deformation parameters.
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.
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.
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.
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.
The well-known Heisenberg–Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [ X , P ] = i ℏ ( 1 + β P 2 ) implies the existence of a minimal length proportional to β . The Bialynicki-Birula–Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β . Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.