# Statistical Mechanics-Based Schrödinger Treatment of Gravity

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## Abstract

**:**

**511**(2018) 139]. In this paper, we revisit this technique with regards to fermions’ behavior (specifically, baryons).

## 1. Introduction

#### 1.1. Emergent Entropy

#### 1.2. Our Goals in Using Schrödinger’s Equation (SE)

#### 1.3. Organizing Our Material

## 2. Quantum Gravitational Potential ${\mathit{E}}_{\mathit{P}}\left(\mathit{r}\right)$ to Be Introduced in the SE

#### 2.1. The Gravitational Potential Function for N Baryons of Mass m

- a and b in the fashion;
- $a={\left(3N\right)}^{\frac{5}{2}}{h}^{3}$;
- $b=32\pi {\left(\pi emK\right)}^{\frac{3}{2}}$, with a total baryons energy K;
- $K={10}^{53}{c}^{2}$ Joules [13].

#### 2.2. A Taylor Approximation (TA) for $V\left(r\right)$

## 3. Exact Solution of the SE

#### 3.1. ${V}_{1}$’s Exact Treatment

**Boundary conditions (BC)**. ${R}_{l}$ must satisfy ${R}_{l}\left({r}_{0}\right)=0$ and ${R}_{l}^{{}^{\prime}}\left({r}_{0}\right)=0$. The two BEs now become

#### 3.2. ${V}_{2}$’s Exact Treatment

#### 3.3. ${V}_{3}$’s Exact Treatment

**Choose $E<0$**

**Choose $E>0$**

#### 3.4. ${V}_{4}$’s Exact Treatment

**Choose $E<0$**

**Choose $E>0$**

## 4. Discussion

- We began by adopting Verlinde’s stance that gravity emerges from an entropy S (entropic force);

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Plastino, A.; Rocca, M.C.
Statistical Mechanics-Based Schrödinger Treatment of Gravity. *Entropy* **2019**, *21*, 682.
https://doi.org/10.3390/e21070682

**AMA Style**

Plastino A, Rocca MC.
Statistical Mechanics-Based Schrödinger Treatment of Gravity. *Entropy*. 2019; 21(7):682.
https://doi.org/10.3390/e21070682

**Chicago/Turabian Style**

Plastino, Angelo, and M. C. Rocca.
2019. "Statistical Mechanics-Based Schrödinger Treatment of Gravity" *Entropy* 21, no. 7: 682.
https://doi.org/10.3390/e21070682