# Fermionic Properties of Two Interacting Bosons in a Two-Dimensional Harmonic Trap

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## Abstract

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## 1. Introduction

## 2. The Hamiltonian

## 3. Analytic Wave Functions to Compare with the Ground State

#### 3.1. First-order Perturbation Theory

#### 3.2. The Non-Interacting Two-Fermion System

#### 3.3. Bosonized Two-fermion System

## 4. The Interaction Effect in the Ground State Energy

#### 4.1. The Energy Contributions

#### 4.2. Exploring the Strongly Interacting Limit for a Short-range Interaction

#### 4.3. Exploring the Short-Range Limit for a Strong Interaction Strength

## 5. The Density Profile and the Two-body Correlations

## 6. Numerical Method

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Ground state energy of two interacting bosons in a two-dimensional harmonic trap, ${E}_{0}$, depending on the interaction strength, g, for a fixed and small interaction range, s. We present the harmonic potential part, $\langle {\mathcal{V}}_{\mathit{ho}}\rangle $, the kinetic part, $\langle \mathcal{K}\rangle $, and the interaction part, $\langle {\mathcal{V}}_{\mathit{int}}\rangle $. The energies were computed numerically using the first $M=200$ single-particle eigenstates of the harmonic oscillator (see Section 6 for details).

**Figure 2.**Ground state energy of two interacting bosons in a harmonic trap, ${E}_{0}$, as in Figure 1, depending on the interaction strength, g, and for a fixed range, s. The exact calculation is compared with the expectation value of the energy of the wave functions of two noninteracting bosons, ${E}_{B}$, two noninteracting fermions, ${E}_{F}$, and the corresponding symmetrized wavefunction, ${E}_{\left|F\right|}$, which are given, respectively, by Equations (8), (11), and (13).

**Figure 3.**Ground state energy of two interacting bosons in a harmonic trap, ${E}_{0}$, depending on the interaction range, s, and for a fixed interaction strength, g. The numerical result is compared with the expectation value of the energy of the wave functions of two noninteracting bosons, ${E}_{B}$, two noninteracting fermions, ${E}_{F}$, and its the symmetrized wavefunction, ${E}_{\left|F\right|}$, which are given, respectively, by Equations (8), (11), and (13).

**Figure 4.**Ground state energy of two interacting bosons in a harmonic trap, ${E}_{0}$, depending on the interaction range, s, for a fixed interaction strength, g. We also show the different contributions, $\langle {\mathcal{V}}_{\mathit{ho}}\rangle $, $\langle \mathcal{K}\rangle $, and ${\langle \mathcal{V}}_{\mathit{int}}\rangle $. The energies were computed numerically using the first $M=200$ single-particle eigenstates of the harmonic oscillator (see Section 6 for details).

**Figure 5.**(

**a**) Density profiles of two noniteracting bosons, ${\rho}_{B}$, two noninteracting fermions, ${\rho}_{F}$, and three numerically computed profiles for different ranges and interaction strength for the interacting two-boson system; (

**b**) Probability of finding a particle at a distance X from the origin once a particle is found at $X=0$ in the same cases. For the numerical calculations, the number of single-particle eigenstates of the harmonic oscillator used was $M=80$ (see Section 6 for details).

**Figure 6.**(

**a**) Density profiles of two noniteracting bosons, ${\rho}_{B}$, two noninteracting fermions, ${\rho}_{F}$, and four numerically computed profiles for different ranges fixing the interaction strength for the interacting two-boson system; (

**b**) Probability of finding a particle at a distance X from the origin once a particle is found at $X=0$ in the same cases. For the numerical calculations the number of single-particle eigenstates of the harmonic oscillator used was $M=80$ (see Section 6 for details).

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**MDPI and ACS Style**

Mujal, P.; Polls, A.; Juliá-Díaz, B.
Fermionic Properties of Two Interacting Bosons in a Two-Dimensional Harmonic Trap. *Condens. Matter* **2018**, *3*, 9.
https://doi.org/10.3390/condmat3010009

**AMA Style**

Mujal P, Polls A, Juliá-Díaz B.
Fermionic Properties of Two Interacting Bosons in a Two-Dimensional Harmonic Trap. *Condensed Matter*. 2018; 3(1):9.
https://doi.org/10.3390/condmat3010009

**Chicago/Turabian Style**

Mujal, Pere, Artur Polls, and Bruno Juliá-Díaz.
2018. "Fermionic Properties of Two Interacting Bosons in a Two-Dimensional Harmonic Trap" *Condensed Matter* 3, no. 1: 9.
https://doi.org/10.3390/condmat3010009