Introduction

Quantum mechanics has unveiled that the vacuum is, in reality, a space brimming with quantum activity, thus challenging the classical understanding of reality [1][2]. This redefined notion of the vacuum is not only crucial for theoretical physics but also has significant practical implications in fields such as cosmology, particle physics, and emerging technologies [2].
In the process of understanding the fundamental yet elusive bases of the universe, the physics of the vacuum emerges as a crucial field of study. This area has undergone significant evolution over time, marking a transition from classical physics to quantum mechanics. Historically, the vacuum was conceived simply as a space of nothingness, a completely empty canvas devoid of matter and energy, a sort of absolute "nothing." However, with the advent of quantum mechanics, this intuitive notion has been radically transformed [1].
In classical physics, the vacuum was seen as a blank canvas, an inert and immutable stage where physical phenomena susceptible to cognition and measurement unfolded. Yet, this understanding began to change in the early 20th century with advances in quantum mechanics. Scientists such as Max Planck, Albert Einstein, Niels Bohr, and many others began to discover that the world at the atomic and subatomic scale did not follow the rules of classical physics and, in some cases, defied Euclidean and mechanistic understanding of reality: it was found that what was known as the “vacuum” is far from being a simple and intuitive “nothing,” instead being a breeding ground for physical phenomena, characterized by quantum fluctuations, virtual photons, antiparticles, and energies that challenge our understanding of reality [2]. Thus, in the quantum framework, the vacuum reveals itself not as an absolute emptiness, but as a turbulent and effervescent environment of activity.
 

Vacuum and the Schrödinger Equation

In the study of the vacuum from the perspective of modern physics, it is essential to understand the fundamentals of quantum mechanics. This understanding begins with the Schrödinger equation, a fundamental formula that provides a mathematical description of the behavior of quantum particles [3].
The Schrödinger equation is inherently important for analyzing and understanding the concept of vacuum in quantum physics. It not only describes the quantum behaviors of particles and systems but also reveals the complex and dynamic nature of the vacuum in the quantum universe.
To describe the motion and behavior of an object in the macroscopic world, Newton's laws are used to predict its behavior. These laws allow for precise calculation of the trajectory, velocity, and position of the object at any given time. However, in the world of very small particles (atomic and subatomic world), "objects" do not behave following Newton's rules. Particles (subatomic particles and elementary particles) at this scale behave in very strange and unpredictable ways. This is where the Schrödinger equation comes into play.
The Schrödinger equation is a set of new rules that redefine Newton's laws to describe how particles move and behave. Instead of specifying exactly where a particle is and how it moves at a given moment, it gives probabilities: it tells us how likely it is to find the particle in a specific place and in a specific state.
To do this, the equation uses the 'wave function'. This wave function is like a map of all the possible positions and energies that the particle can have. However, unlike a conventional map used to navigate cities, this map gives us probabilities. The more intense the "wave" in a part of the map, the more likely we are to find the particle there. The Schrödinger equation is generally expressed in two forms: the time-dependent form and the time-independent form. The time-dependent form describes how the quantum state of a system changes over time, given by the equation: