Spectrum and eigenfunctions in the momentum representation for 1D Coulomb potential with deformed Heisenberg algebra leading to minimal length are found exactly. It is shown that correction due to the deformation is proportional to square root of the deformation parameter. We obtain the same spectrum using Bohr-Sommerfeld quantization condition.
We present a definition of the two-sided inverse of position operator in general case of deformed Heisenberg algebra leading to minimal length. Energy spectrum and eigenfunctions in momentum space for 1D Coulomb-like potential in deformed space are found exactly. We analyse the energy spectrum for different partial cases of deformation function and find that correction due to the deformation highly depends on type of the deformation function.
The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Ke…
In general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the $δ'(x)$ potential as a linear kernel of potential energy operator in momentum representation. We find exactly the energy level and corresponding eigenfunction for $δ'(x)$ and $δ(x)-δ'(x)$ potentials in deformed space with arbitrary function of deformation. The energy spectrum for different partial cases of deformation function is analysed.
Corrections to the ns-levels of hydrogen atom in deformed space with minimal length
研究了形变海森堡代数中的氢原子问题，得到了最小长度的存在性。利用我们在前人工作中发展起来的修正的微扰理论[M. M. Stetsko和V.M. Tkachuk，Phys.Rev.A74，012101（2006）]，我们计算了对氢原子的任意s能级的修正。我们得到了一个简单的关系，估计的最小长度。我们还比较了这里得到的最小长度的估计与在前面的调查中得到的结果。
We reduce two-body problem to the one-body problem in general case of deformed Heisenberg algebra leading to minimal length.Two-body problems with delta and Coulomb-like interactions are solved exactly. We obtain analytical expression for the energy spectrum for partial cases of deformation function. The dependence of the energy spectrum on the center-of-mass momentum is found. For special case of deformation function, which correspondes to cutoff procedure in momentum space it is shown that this dependence is more likely to observe for identical particles.
In general case of deformed Heisenberg algebra leading to the minimal length we present a definition of the square inverse position operator. Our proposal is based on the functional analysis of the square position operator. Using this definition a particle in the field of the square inverse position potential is studied. We have obtained analytical and numerical solutions for the energy spectrum of the considerable problem in different cases of deformation function. We find that the energy spectrum slightly depends on the choice of deformation function.
$1+1$-dimensional Dirac oscillator with minimal uncertainty in position and maximal in momentum is investigated. To obtain energy spectrum SUSY QM technique is applied. It is shown that the Dirac oscillator has two branches of spectrum, the first one gives the standard spectrum of the Dirac oscillator when the parameter of deformation goes to zero and the second branch does not have nondeformed limit. Maximal momentum brings an upper bound for the energy and it gives rise to the conclusion that the energy spectrum contains a finite number of eigenvalues. We also calculate partition function fo…
We consider the elastic scattering in deformed space with minimal length. We give the basic relation for the elastic scattering in deformed space. We also investigate the partial wave method in deformed space. It is shown that the relations for the scattering amplitude and cross-section formally coincides with ordinary ones.
A one-dimensional deformed Heisenberg algebra $[X,P]=if(P)$ is studied. We answer the question: For what function of deformation $f(P)$ there exists a nonzero minimal uncertainty in position (minimal length). We also find an explicit expression for the minimal length in the case of arbitrary function of deformation.
Studies in string theory and quantum gravity lead to the Generalized Uncertainty Principle (GUP) and suggest the existence of a fundamental minimal length which, as was established, can be obtained within the deformed Heisenberg algebra. The first look on the classical motion of bodies in a space with corresponding deformed Poisson brackets in a uniform gravitational field can give an impression that bodies of different mass fall in different ways and thus the equivalence principle is violated. Analyzing the kinetic energy of a composite body we find that the motion of its center of mass in th…
We review the essentials of the formalism of quantum mechanics based on a deformed Heisenbeg algebra, leading to the existence of a minimal length scale. We compute in this context, the energy spectra of the pseudoharmonic oscillator and Kratzer potentials by using a perturbative approach. We derive the molecular constants, which characterize the vibration--rotation energy levels of diatomic molecules, and investigate the effect of the minimal length on each of these parameters for both potentials. We confront our result to experimental data for the hydrogen molecule to estimate an order of ma…
Entropic entanglement measures of a two-dimensional system of two Coulombically interacting particles confined in an anisotropic harmonic potential are discussed in dependence on the anisotropy and the interaction strength. The harmonic approximation appears exact in the strong interaction limit, allowing determination of the asymptotic expression for the linear entropy. Entanglement properties are dramatically influenced by the anisotropy of the confining potential in the strong-correlation regime.
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 every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the mini…
We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be constructed explicitly, we derive the commutation relations for the dynamical variables in noncommutative space-time. We compute minimal areas resulting from these relations, i.e. finitely extended regions for which it is impossible to resolve any substructure in form of measurable knowledge. The size of the regions we find is determined by the noncommutativ…
We have obtained the uncertainty relations for arbitrary states of the hydrogen atom. It is shown that the minimal value of the uncertainty relation is attained for the circular Rydberg states.
A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schroëdinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator do not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is sol…
For composite systems made of $N$ different particles living in a space characterized by the same deformed Heisenberg algebra, but with different deformation parameters, we define the total momentum and the center-of-mass position to first order in the deformation parameters. Such operators satisfy the deformed algebra with new effective deformation parameters. As a consequence, a two-particle system can be reduced to a one-particle problem for the internal motion. As an example, the correction to the hydrogen atom $n$S energy levels is re-evaluated. Comparison with high-precision experimental…
Perivolaropoulos has recently proposed a position-deformed Heisenberg algebra which includes a maximal length [Phys.Rev.95, 103523 (2017)]. He has shown that this length scale naturally emerges in the context of cosmological particle's horizon or cosmic topology. Following this work, we propose a new deformed algebra and derive the maximal length uncertainty and its corresponding minimal momentum uncertainty from the generalized uncertainty principle. We also construct the corresponding Fourier transform and its inverse representations. Finally, we propose n-dimensional representation of this …
We study the Klein-Gordon equation for Coulomb potential, V(r)=(-Ze^{2})/r, in quantum mechanics with a minimal length. The zero energy solution is obtained analytically in momentum space in terms of Heun's functions. The asymptotic behavior of the solution shows that the presence of a minimal length regularize the potential in the strong attractive regime, Z>68. The equation with nonzero energy is established in a particular case in the first order of the deformation parameter; it is a generalized Heun's equation.
We derive continuity equation and exact expression for flow probability density in a space with arbitrary deformed algebra leading to minimal length. In coordinate representation the flow probability density is presented as infinite series over parameter of deformation which in momentum representation can be casted into exact closed form determined by deformed kinetic energy. The flow probability density is calculated explicitly for plane wave and for superposition of two plane waves.
A minimal observable length is a common feature of theories that aim to merge quantum physics and gravity. Quantum mechanically, this concept is associated to a nonzero minimal uncertainty in position measurements, which is encoded in deformed commutation relations. In spite of increasing theoretical interest, the subject suffers from the complete lack of dedicated experiments and bounds to the deformation parameters are roughly extrapolated from indirect measurements. As recently proposed, low-energy mechanical oscillators could allow to reveal the effect of a modified commutator. Here we ana…
A construction method of infinite families of quasi-exactly solvable position-dependent mass Schrödinger equations with known ground and first excited states is proposed in a deformed supersymmetric background. Such families correspond to extensions of known potentials endowed with a deformed shape invariance property. Two different approaches are combined. The first one is a generating function method, which enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential…
Minimal nonintegrable models with three-site interactions
School of Physics, Northwest University（西北大学物理学院）；Institute of Modern Physics, Northwest University（西北大学现代物理研究所）；Shaanxi Key Laboratory for Theoretical Physics Frontiers（陕西理论物理前沿重点实验室）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论中心）；Fundamental Discipline Research Center for Quantum Science and Technology of Shaanxi Province（陕西省量子科学与技术基础学科研究中心）
The relativistic quantum dynamics of scalar bosons in the background of a full vector coupling (minimal plus nonminimal vector couplings) is explored in the context of the Duffin-Kemmer-Petiau formalism. The Coulomb phase shift is determined for a general mixing of couplings and it is shown that the space component of the nonminimal coupling is a {\it sine qua non} condition for the exact closed-form scattering amplitude. It follows that the Rutherford cross section vanishes in the absence of the time component of the minimal coupling. Bound-state solutions obtained from the poles of the parti…
