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## Minimal length effect on the broadening of free wave-packets and its physical implications

## Abstract

We study the generalized uncertainty principle (GUP) modified time evolution for the width of wave packets for a scalar potential. The free particle case is solved exactly where the wave-packet broadening is modified by a coupling between the GUP parameter and higher-order moments in the probability distribution in momentum space. We consider two popular forms of deformations widely used in the literature—one of which modifies the commutator with a quadratic term in momentum, while the other modifies it with terms both linear and quadratic in momentum. Unlike the standard case, satisfying Heisenberg uncertainty, here the GUP modified broadening rates, for both deformations, not only depend on the initial size (in both position and momentum space) of the wave packet but also on the initial probability distribution and momentum of the particle. The new rates of wave-packet broadening, for both situations, are modified by a handful of new terms—such as the skewness and kurtosis coefficients, as well as the (constant) momentum of the particle. Comparisons with the standard Heisenberg uncertainty principle–based results show potentially measurable differences in the rates of free wave-packet broadening for physical systems such as the C60 and C176 molecules, and more so for large organic molecular wavemore »

- Authors:

- Fermilab
- Colima U.

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1523427

- Report Number(s):
- arXiv:1812.06112; FERMILAB-PUB-18-704

1709339

- Grant/Contract Number:
- AC02-07CH11359

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Phys.Rev.

- Additional Journal Information:
- Journal Volume: D100; Journal Issue: 2

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Villalpando, Carlos, and Modak, Sujoy K. Minimal length effect on the broadening of free wave-packets and its physical implications. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.024054.
```

```
Villalpando, Carlos, & Modak, Sujoy K. Minimal length effect on the broadening of free wave-packets and its physical implications. United States. doi:10.1103/PhysRevD.100.024054.
```

```
Villalpando, Carlos, and Modak, Sujoy K. Sat .
"Minimal length effect on the broadening of free wave-packets and its physical implications". United States. doi:10.1103/PhysRevD.100.024054.
```

```
@article{osti_1523427,
```

title = {Minimal length effect on the broadening of free wave-packets and its physical implications},

author = {Villalpando, Carlos and Modak, Sujoy K.},

abstractNote = {We study the generalized uncertainty principle (GUP) modified time evolution for the width of wave packets for a scalar potential. The free particle case is solved exactly where the wave-packet broadening is modified by a coupling between the GUP parameter and higher-order moments in the probability distribution in momentum space. We consider two popular forms of deformations widely used in the literature—one of which modifies the commutator with a quadratic term in momentum, while the other modifies it with terms both linear and quadratic in momentum. Unlike the standard case, satisfying Heisenberg uncertainty, here the GUP modified broadening rates, for both deformations, not only depend on the initial size (in both position and momentum space) of the wave packet but also on the initial probability distribution and momentum of the particle. The new rates of wave-packet broadening, for both situations, are modified by a handful of new terms—such as the skewness and kurtosis coefficients, as well as the (constant) momentum of the particle. Comparisons with the standard Heisenberg uncertainty principle–based results show potentially measurable differences in the rates of free wave-packet broadening for physical systems such as the C60 and C176 molecules, and more so for large organic molecular wave packets. In doing so, we open a path to scan the GUP parameter space by several orders of magnitude inside the best existing upper bounds for both forms of GUP.},

doi = {10.1103/PhysRevD.100.024054},

journal = {Phys.Rev.},

number = 2,

volume = D100,

place = {United States},

year = {2019},

month = {7}

}

Works referenced in this record:

##
Size analysis of single fullerene molecules by electron microscopy

journal, January 2004

- Goel, Anish; Howard, Jack B.; Vander Sande, John B.
- Carbon, Vol. 42, Issue 10, p. 1907-1915