The generalized uncertainty principle with commuting coordinates

2019 ◽  
Vol 34 (33) ◽  
pp. 1950267
Author(s):  
Won Sang Chung
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Generalized Uncertainty Principle ◽  
Three Dimensional ◽  
One Dimension ◽  
Momentum Representation ◽  
Position Representation

In this paper, we present a new D-dimensional generalized uncertainty principle (GUP) algebra with commuting coordinates which recovers [Formula: see text] in one dimension. We find two representations for this GUP: momentum representation and position representation. We discuss the GUP-corrected three-dimensional quantum mechanics in position representation for a small [Formula: see text]. Finally, we discuss the momentum wave function and GUP-corrected Fermi metal theory.

scholarly journals On the Nature of the Change in the Wave Function in a Measurement in Quantum Mechanics

2020 ◽  
Author(s):  
Douglas Michael Snyder
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Empirical Work ◽  
Physical Review ◽  
Physical Interaction ◽  
Measuring Instrument ◽  
Physical Entity

Generally a central role has been assigned to an unavoidable physical interaction between the measuring instrument and the physical entity measured in the change in the wave function that often occurs in measurement in quantum mechanics. A survey of textbooks on quantum mechanics by authors such as Dicke and Witke (1960), Eisberg and Resnick (1985), Gasiorowicz (1974), Goswami (1992), and Liboff (1993) supports this point. Furthermore, in line with the view of Bohr and Feynman, generally the unavoidable interaction between a measuring instrument and the physical entity measured is considered responsible for the uncertainty principle. A gedankenexperiment using Feynman's double-hole interference scenario shows that physical interaction is not necessary to effect the change in the wave function that occurs in measurement in quantum mechanics. Instead, the general case is that knowledge is linked to the change in the wave function, not a physical interaction between the physical existent measured and the measuring instrument. Empirical work on electron shelving that involves null measurements, or what Renninger called negative observations (Zeitschrift fur Physik, vol. 158, p. 417), supports these points. Work on electron shelving is reported by Dehmelt and his colleagues (Physical Review Letters, vol. 56, p. 2797), Wineland and his colleagues (Physical Review Letters, vol. 57, p. 1699), and Sauter, Neuhauser, Blatt, and Toschek (Physical Review Letters, vol. 57, p. 1696). Originally appeared in arXiv on December 19, 1995. It can be accessed at arXiv:quant-ph/9601006 .

Three-dimensional quantum mechanics in a curved space based on the q-addition

2019 ◽  
Vol 34 (29) ◽  
pp. 1950177
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Coordinate System ◽  
Three Dimensional ◽  
One Dimension ◽  
Cartesian Coordinate System ◽  
Spherical Coordinate ◽  
Curved Space ◽  
Dimension Formula ◽  
Cartesian Coordinate

In this paper, we extend the theory of the [Formula: see text]-deformed quantum mechanics in one dimension[Formula: see text] into three-dimensional case. We relate the [Formula: see text]-deformed quantum theory to the quantum theory in a curved space. We discuss the diagonal metric based on [Formula: see text]-addition in the Cartesian coordinate system and core radius of neutron star. We also discuss the diagonal metric based on [Formula: see text]-addition in the spherical coordinate system and [Formula: see text]-deformed Heisenberg atom model.

scholarly journals A CLASS OF GUP SOLUTIONS IN DEFORMED QUANTUM MECHANICS

2010 ◽  
Vol 19 (12) ◽  
pp. 2003-2009 ◽  
Author(s):  
POURIA PEDRAM
Keyword(s):  
Black Hole ◽  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Uncertainty Principle ◽  
Loop Quantum Gravity ◽  
Black Hole Physics ◽  
Generalized Uncertainty Principle ◽  
Heisenberg Uncertainty Principle ◽  
Deformed Algebra ◽  
Heisenberg Uncertainty

Various candidates of quantum gravity such as string theory, loop quantum gravity and black hole physics all predict the existence of a minimum observable length which modifies the Heisenberg uncertainty principle to the so-called generalized uncertainty principle (GUP). This approach results from the modification of the commutation relations and changes all Hamiltonians in quantum mechanics. In this paper, we present a class of physically acceptable solutions for a general commutation relation without directly solving the corresponding generalized Schrödinger equations. These solutions satisfy the boundary conditions and exhibit the effect of the deformed algebra on the energy spectrum. We show that this procedure prevents us from doing equivalent but lengthy calculations.

scholarly journals Quantum cosmology with dynamical vacuum in a minimal-length scenario

2021 ◽  
Vol 81 (4) ◽  
Author(s):  
M. F. Gusson ◽  
A. Oakes O. Gonçalves ◽  
R. G. Furtado ◽  
J. C. Fabris ◽  
J. A. Nogueira
Keyword(s):  
Wave Function ◽  
Commutation Relation ◽  
Uncertainty Principle ◽  
Quantum Cosmology ◽  
Generalized Uncertainty Principle ◽  
Minimal Length ◽  
Minimum Length ◽  
Position Space ◽  
Infinite Norm ◽  
The Universe

AbstractIn this work, we consider effects of the dynamical vacuum in quantum cosmology in presence of a minimum length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation $$[{\hat{X}},{\hat{P}}] := \frac{i\hbar }{ 1 - \beta {\hat{P}}^2 }$$ [ X ^ , P ^ ] : = i ħ 1 - β P ^ 2 . We determine the wave function of the Universe $$ \psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) , which is solution of the modified Wheeler–DeWitt equation in the representation of the quasi-position space, in the limit where the scale factor of the Universe is small. Although $$\psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) is a physically acceptable state it is not a realizable state of the Universe because $$ \psi _{qp}(\xi ,t)$$ ψ qp ( ξ , t ) has infinite norm, as in the ordinary case with no minimal length.

scholarly journals Possible Unification of Quantum Mechanics and General Relativity Theory Based on the Three-Dimensional Quantized Spaces

2018 ◽  
Author(s):  
Jae-Kwang Hwang
Keyword(s):  
General Relativity ◽  
Wave Function ◽  
Quantum Mechanics ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Energy Transition ◽  
Space Time ◽  
Relativity Theory ◽  
General Relativity Theory ◽  
Dimensional Space Time

Three-dimensional quantized space model is newly introduced. Quantum mechanics and relativity theory are explained in terms of the warped three-dimensional quantized spaces with the quantum time width (Dt=tq). The energy is newly defined as the 4-dimensional space-time volume of E = cDtDV in the present work. It is shown that the wave function of the quantum mechanics is closely related to the warped quantized space shape with the space time-volume. The quantum entanglement and quantum wave function collapse are explained additionally. The special relativity theory is separated into the energy transition associated with the space-time shape transition of the matter and the momentum transition associated with the space-time location transition. Then, the quantum mechanics and the general relativity theory are about the 4-dimensional space-time volume and the 4-dimensional space-time distance, respectively.

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