# A Study of Strong Confinement Regions Using Informational Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. System of Interest

#### 2.2. Shannon Informational Entropy

## 3. Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Values of $\Delta {S}_{r}$ in different quantum states as a function of ${r}_{c}$ on a logarithmic scale for the (

**a**) hydrogen confined, (

**b**) ion-sphere, and (

**c**) confined harmonic oscillator.

**Figure 2.**Values of $\Delta {S}_{p}$ in different quantum states as a function of ${r}_{c}$ on a logarithmic scale for the (

**a**) hydrogen confined, (

**b**) ion-sphere, and (

**c**) confined harmonic oscillator.

**Figure 3.**Values of $\Delta {S}_{t}$ in different quantum states as a function of ${r}_{c}$ on a logarithmic scale for the (

**a**) hydrogen confined, (

**b**) ion-sphere, and (

**c**) confined harmonic oscillator.

**Figure 4.**Values of $\Delta {S}_{r}$ in ground state as a function of ${r}_{c}$ for the hydrogen confined (HC), ion-sphere (IS), and confined harmonic oscillator (HO). The vertical lines delimit the region of strong confinement for each system.

**Figure 5.**Values of ${S}_{t}$ in different quantum states as a function of ${r}_{c}$ on a logarithmic scale in the weak and intermediate regions for the (

**a**) hydrogen confined, (

**b**) ion-sphere, (

**c**) confined harmonic oscillator, and (

**d**) particle confined in a cage.

**Figure 6.**Values of ${S}_{r}$ and ${S}_{p}$ in different quantum states as a function of ${r}_{c}$ on a logarithmic scale for the particle confined in a cage.

**Figure 7.**Values of ${S}_{r}$ in ground state as a function of ${r}_{c}$ on logarithmic scale for the hydrogen confined (HC), ion-sphere (IS), confined harmonic oscillator (HO), and particle confined (eC).

**Table 1.**Confinement regions for the hydrogen confined (HC), ion-sphere (IS), and confined harmonic oscillator (HO).

$\mathit{\sigma}$ | Strong | Intermediate | Weak | |
---|---|---|---|---|

HC | 0.040 | ${r}_{c}<1.17$ | $1.17\le {r}_{c}\le 4.51$ | ${r}_{c}>4.51$ |

IS | 0.035 | ${r}_{c}<0.56$ | $0.56\le {r}_{c}\le 2.36$ | ${r}_{c}>2.36$ |

HO | 0.000 | ${r}_{c}<2.00{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ | – | – |

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**MDPI and ACS Style**

Santos, A.d.J.; Prudente, F.V.; Guimarães, M.N.; Nascimento, W.S.
A Study of Strong Confinement Regions Using Informational Entropy. *Quantum Rep.* **2022**, *4*, 544-557.
https://doi.org/10.3390/quantum4040039

**AMA Style**

Santos AdJ, Prudente FV, Guimarães MN, Nascimento WS.
A Study of Strong Confinement Regions Using Informational Entropy. *Quantum Reports*. 2022; 4(4):544-557.
https://doi.org/10.3390/quantum4040039

**Chicago/Turabian Style**

Santos, Ademir de J., Frederico V. Prudente, Marcilio N. Guimarães, and Wallas S. Nascimento.
2022. "A Study of Strong Confinement Regions Using Informational Entropy" *Quantum Reports* 4, no. 4: 544-557.
https://doi.org/10.3390/quantum4040039