# On entropic uncertainty relations in the presence of a minimal length

## Abstract

Entropic uncertainty relations for the position and momentum within the generalized uncertainty principle are examined. Studies of this principle are motivated by the existence of a minimal observable length. Then the position and momentum operators satisfy the modified commutation relation, for which more than one algebraic representation is known. One of them is described by auxiliary momentum so that the momentum and coordinate wave functions are connected by the Fourier transform. However, the probability density functions of the physically true and auxiliary momenta are different. As the corresponding entropies differ, known entropic uncertainty relations are changed. Using differential Shannon entropies, we give a state-dependent formulation with correction term. State-independent uncertainty relations are obtained in terms of the Rényi entropies and the Tsallis entropies with binning. Such relations allow one to take into account a finiteness of measurement resolution. - Highlights: • Entropic uncertainty bounds increase for position and physically true momentum. • Correction terms are always nonzero for wave packets of a finite width. • An increase of uncertainty bounds depends on acceptance functions of apparatuses.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22701508

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 382; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMMUTATION RELATIONS; ENTROPY; FOURIER TRANSFORMATION; PROBABILITY DENSITY FUNCTIONS; UNCERTAINTY PRINCIPLE; WAVE FUNCTIONS; WAVE PACKETS

### Citation Formats

```
Rastegin, Alexey E., E-mail: rast@api.isu.ru.
```*On entropic uncertainty relations in the presence of a minimal length*. United States: N. p., 2017.
Web. doi:10.1016/J.AOP.2017.04.014.

```
Rastegin, Alexey E., E-mail: rast@api.isu.ru.
```*On entropic uncertainty relations in the presence of a minimal length*. United States. doi:10.1016/J.AOP.2017.04.014.

```
Rastegin, Alexey E., E-mail: rast@api.isu.ru. Sat .
"On entropic uncertainty relations in the presence of a minimal length". United States. doi:10.1016/J.AOP.2017.04.014.
```

```
@article{osti_22701508,
```

title = {On entropic uncertainty relations in the presence of a minimal length},

author = {Rastegin, Alexey E., E-mail: rast@api.isu.ru},

abstractNote = {Entropic uncertainty relations for the position and momentum within the generalized uncertainty principle are examined. Studies of this principle are motivated by the existence of a minimal observable length. Then the position and momentum operators satisfy the modified commutation relation, for which more than one algebraic representation is known. One of them is described by auxiliary momentum so that the momentum and coordinate wave functions are connected by the Fourier transform. However, the probability density functions of the physically true and auxiliary momenta are different. As the corresponding entropies differ, known entropic uncertainty relations are changed. Using differential Shannon entropies, we give a state-dependent formulation with correction term. State-independent uncertainty relations are obtained in terms of the Rényi entropies and the Tsallis entropies with binning. Such relations allow one to take into account a finiteness of measurement resolution. - Highlights: • Entropic uncertainty bounds increase for position and physically true momentum. • Correction terms are always nonzero for wave packets of a finite width. • An increase of uncertainty bounds depends on acceptance functions of apparatuses.},

doi = {10.1016/J.AOP.2017.04.014},

journal = {Annals of Physics},

issn = {0003-4916},

number = ,

volume = 382,

place = {United States},

year = {2017},

month = {7}

}