# The Vortex Surface in a Three-Body Quantum System

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## Abstract

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## 1. Introduction

## 2. A World of Vortices

#### 2.1. Vortex Atom

#### 2.2. Vortices in Quantum Few-Body Systems

## 3. Quantum Vortex

## 4. Quantum Vortex in Atomic Collisions

**v**being the velocity and

**b**being the impact parameter. While convincingly demonstrating the emergence and evolution of quantum vortices in collision processes, and even that some of these vortices can stabilize and remain even at very large times and macroscopic distances, these studies did not indicate how it would be possible to detect their presence in the corresponding cross sections. To establish this link, we must apply the following theorem.

#### 4.1. Imaging Theorem

#### 4.2. Reduction in the Number of Variables with Conservation and Symmetry Laws

#### 4.2.1. Rochester Geometry

#### 4.2.2. Harvard Geometry

#### 4.3. Restrictive Geometries

## 5. Vortex Points in Simple Impact Ionization of Charged Particles

#### 5.1. Electron Impact

#### 5.2. Positron Impact

## 6. Vortex Lines

## 7. Vortex Rings

## 8. Vortex Surface

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**In a rotational vortex (

**left**), each fluid element rotates on itself, like what happens in a rigid body. In an irrotational or free vortex (

**right**), each fluid element maintains its direction fixed in space.

**Figure 2.**The Rochester geometry uses the polar angles ${\theta}_{p}$ y ${\theta}_{e}$ of two particles, the azimuthal angle $\varphi $ between them, and one energy, ${E}_{p}={k}_{p}^{2}/2{m}_{p}$ or ${E}_{e}={k}_{e}^{2}/2{m}_{e}$, as the four variables necessary to fully represent the final state of the three-body system, attending to the conditions of conservation and symmetry of the collision process.

**Figure 3.**The Harvard geometry uses the angle $\theta $ between the initial direction of the projectile and the plane formed with the final momentum of the particles $p$ and $e$, the angles ${\xi}_{p}$ and ${\xi}_{e}$ on that plane, and the energy of one of them (i.e., ${E}_{p}$ or ${E}_{p}$).

**Figure 4.**Square modulus of the transition matrix ${\left|T\right|}^{2}$, for the ionization of atomic hydrogen by the impact of positrons of 275 eV. A collinear geometry is used, where the electron and positron exit in the same direction. ${k}_{\parallel}$ y ${k}_{\perp}$ are their momentum components parallel and perpendicular to the initial velocity of positron $\mathbf{v}$, respectively. Both components are normalized by means of the maximum value ${k}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\sqrt{{m}_{e}^{2}{\mathrm{v}}^{2}-2\epsilon}$, which can reach the electron in an ionization process. The arrows represent the directions of the generalized velocity field $\mathbf{u}=\mathrm{I}\mathrm{m}{\nabla}_{\mathit{k}}\mathrm{ln}T$.

**Figure 5.**Vortex line for the ionization of atomic hydrogen by positron impact of 275 eV [36]. A coplanar geometry is used, where the initial velocity of the projectile and the final moments of the electron and positron are in the same plane. This corresponds to fixing $\varphi =0$ in Rochester geometry. As in the figure above, ${k}_{\parallel}$ y ${k}_{\perp}$ are the components of the momentum of the electron parallel and perpendicular to the initial velocity of the positron $\mathbf{v}$, respectively. Both components are normalized by means of the maximum value ${k}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\sqrt{{m}_{e}^{2}{\mathrm{v}}^{2}-2\epsilon}$, which can reach the electron in an ionization process. The vertical axis represents the relative angle ${\theta}_{p}-{\theta}_{e}$ between the exit directions of the positron and the electron, expressed in radians. In the plane corresponding to ${\theta}_{p}={\theta}_{e}$, the square modulus of the transition matrix ${\left|T\right|}^{2}$ shown in Figure 4 is displayed.

**Figure 6.**Same as in Figure 5. The vertical axis represents the perpendicular component of the positron’s final moment.

**Figure 7.**Vortex rings in the FDCS for the ionization of atomic hydrogen by impact of positrons of 100 eV in the space of the momentum $\mathit{k}$ of the electron (in atomic units), when the positron does not vary its direction of movement ($\theta =0$).

**Figure 8.**Vortex lines in the FDCS for the ionization of atomic hydrogen by impact of positrons of 100 eV in the space of momentum of the emitted electron (in atomic units). The positron is deflected at an angle of $\theta =5\xb0$ in the plane ${k}_{y}=0$.

**Figure 11.**Vortex surface in the FDCS for the ionization of atomic hydrogen by impact of positrons of 100 eV in the space of momentum of the emitted electron. The positron is dispersed in the plane ${k}_{y}=0$. Each figure shows a view of the vortex rotated approximately 45 degrees clockwise from the previous one, starting with the top-left figure, and every line represents a cut of the surface corresponding to a value other than the angle $\theta $ of deviation of the projectile. For simplicity, only a surface vortex fragment corresponding to the range is displayed, $0\xb0\le \theta \le 25\xb0$.

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**MDPI and ACS Style**

Guarda, T.A.; Navarrete, F.; Barrachina, R.O.
The Vortex Surface in a Three-Body Quantum System. *Atoms* **2023**, *11*, 147.
https://doi.org/10.3390/atoms11110147

**AMA Style**

Guarda TA, Navarrete F, Barrachina RO.
The Vortex Surface in a Three-Body Quantum System. *Atoms*. 2023; 11(11):147.
https://doi.org/10.3390/atoms11110147

**Chicago/Turabian Style**

Guarda, Tamara A., Francisco Navarrete, and Raúl O. Barrachina.
2023. "The Vortex Surface in a Three-Body Quantum System" *Atoms* 11, no. 11: 147.
https://doi.org/10.3390/atoms11110147