Deformation Of Quantum Mechanics Research Articles

QUANTUM MECHANICS AT PLANCK'S SCALE AND DENSITY MATRIX

In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies in the fact that here the density matrix are subjected to deformation, whereas in the previous approaches only commutators are deformed. The density matrix obtained by deforming the quantum-mechanical one is named the density pro-matrix throughout this paper. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation. The proposed approach allows a description of the dynamics. In particular, the explicit form of the deformed Liouville's equation and the deformed Shrödinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that the results obtained here allow one to deduce in a simple and natural way the Bekenstein–Hawking's formula for black hole entropy in semiclassical approximation.

Deformation of quantum mechanics in fractional-dimensional space

A new kind of deformed calculus (the D-deformed calculus) that takes place in fractional-dimensional spaces is presented. The D-deformed calculus is shown to be an appropriate tool for treating fractional-dimensional systems in a simple way and quite analogous to their corresponding one-dimensional partners. Two simple systems, the free particle and the harmonic oscillator in fractional- dimensional spaces are reconsidered into the framework of the D-deformed quantum mechanics. Confined states in a D-deformed quantum well are studied. D-deformed coherent states are also found.

A (p,q)-Deformation of Quantum Mechanics**The project supported by Natural Science Foundation of Jiangxi Province of China

On a one-dimensional lattice with a geometric sequence spacing, the Hermitian conjugation of a (p,q)-derivative operator is discussed by means of (p,q)-integration. Then a (p,q)-deformation of both the Heisenberg algebra for the canonical coordinates and the Heisenberg–Weyl algebra for the harmonic oscillator is presented. It is shown that although in the algebraic aspect the (p,q)-deformation discussed here is identical with q-deformation given by Truong, the (p,q)-deformed Schrödinger picture is in fact different from the q-deformed one.

Isorepresentations of the Lie-Isotopic SU(2) Algebra with Applications to Nuclear Physics and to Local Realism

In this note, we study the nonlinear-nonlocal-noncanonical, axiom-preserving isotopies/Q-operator deformations SUQ(2) of the SU(2) spin-isospin symmetry. We prove the local isomorphism SUQ(2)≈SU(2), construct and classify the isorepresentations of SUQ(2), identify the emerging generalizations of Pauli matrices, and show their lack of unitary equivalence to the conventional representations. The theory is applied for the reconstruction of the exact SU(2)-isospin symmetry in nuclear physics with equal p and n masses in isospaces. We also prove that Bell's inequality and the von Neumann theorem are inapplicable under isotopies, thus permitting the isotopic completion/Q-operator deformation of quantum mechanics studied in this note which is considerably along the celebrated argument by Einstein, Podolsky and Rosen.

Comment on the deformation of quantum mechanics

For original paper see Li et al., ibid, vol.25, p.6779 (1992) In this comment we point out a serious mistake in a recent paper by Li and Sheng and present the correct version for q-deformed quantum mechanics.

QUANTUM DEFORMATIONS OF QUANTUM MECHANICS

Based on a deformation of the quantum mechanical phase space we study q-deformations of quantum mechanics for qk=1 and 0<q<1. After defining a q-analog of the scalar product on the function space we discuss and compare the time evolution of operators in both cases. A formulation of quantum mechanics for qk=1 is given and the dynamics for the free Hamiltonian is studied. For 0<q<1 we develop a deformation of quantum mechanics and the cases of the free Hamiltonian and the one with a x2-potential are solved in terms of basic hypergeometric functions.

A deformation of quantum mechanics

On a one-dimensional lattice without uniform interval, the Hermitian conjugation of a q-differential operator is discussed. Then a deformation of quantum mechanics in one dimension is presented. As an application, the harmonic oscillator is discussed. The energy spectrum and the eigenfunctions are shown to depend on an arbitrary deformation function. The deformed coherent states are also discussed. It is found that the completeness relation of coherent states holds for the case of q-coherent states, i.e. the deformation of the Heisenberg-Weyl algebra is a q-analogue Hopf algebra.