The end of the nineteenth century marked the apotheosis of the mechanistic worldview. The universe appeared to be a perfect clockwork mechanism, every motion strictly determined by precise laws. Newton gave the world the formulas of motion, Maxwell — the equations of light and electromagnetism. Laplace confidently declared that if one knew all the forces and positions of bodies, one could calculate both the future and the past down to the smallest detail.
It seemed that mystery itself had been banished from the world — only problems remained, awaiting computation. Physics described what is, leaving no room for wonder. Yet within the very core of that confidence, doubt was already beginning to grow.
Paradoxically, the first blow to the mechanistic picture came from James Clerk Maxwell himself. His equations revealed that light is not a stream of particles but a disturbance in an invisible field pervading space. Thus, physics for the first time encountered the concept of a field — not a thing, but a state; not an object, but a structure of relations. The world ceased to be a collection of bodies and began to resemble a fabric, where changes ripple through as waves.
This discovery prepared the ground for a coming revolution: from mechanics of bodies to physics of interactions, from rigid forms to a world of oscillations, probabilities, and meanings.
When in the twentieth century light turned out to be both a wave and a particle, and the electron a being without a definite trajectory, the ground beneath science truly shook. Space and time lost their absoluteness. Even the observer ceased to be “outside the system” — nature seemed to refuse the role of a passive witness and began to speak in its own, unfamiliar language. Determinism was collapsing: the universe was no longer a machine but a living fabric of uncertainty.
Out of this trembling was born the twentieth century — the century in which physics for the first time looked directly at possibility itself. And there, amid this intellectual earthquake, appeared Erwin Schrödinger — the man who, with a single equation, translated the language of science from the description of facts to the description of becoming.
For the first time, instead of a trajectory came the wave function — not a line, but a cloud of probabilities. It does not tell us where the particle is, but everything it could be, as long as no one looks.
Thus began a new era: physics ceased to be the science of bodies and became the science of potentials — of hidden states from which the observable emerges. Schrödinger, perhaps without realizing it, founded a philosophy in which possibility itself becomes the source of reality.
Contents
What Is the Wave Function
The formula that opened this new reality looks deceptively simple:
\( i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi \)
It reads astonishingly clearly: the evolution of the wave state over time is determined by the Hamiltonian (\(\hat{H}\)) — an operator containing all information about the system’s energy. The Hamiltonian is the heart of the system, its semantic structure that dictates how potential unfolds in time. Possibility does not flow arbitrarily — it follows the inner logic of energy.
This equation made time itself not just a parameter but a living process of becoming — it wove the flow of time into the fabric of probability. Yet beneath those dry symbols hides a revolution: a passage from matter to meaning, from the description of what has happened to the description of what may happen.
The formula placed at the center not the thing itself, but the mode of its manifestation. The protagonist became \(\psi\) — a mysterious function whose change over time determines the evolution of the system. It contains all information about what the system may become, though it has not yet done so.
At first, \(\psi\) was seen as a mere computational tool — an auxiliary mathematical quantity without physical reality. But gradually it became clear that behind these abstract amplitudes lies the world itself in its potential form, before any fact is fixed.
The Schrödinger wave function is not an object and not a particle. It is a semantic field of probabilities, describing every possible way a quantum system might manifest itself in interaction with an observer. It can represent position, momentum, energy, spin — all depending on what question we ask nature, and in which basis we choose to view the state.
To connect this abstraction with observable facts, one takes the square of its modulus — \(|\psi|^2\). This expression gives the probability of finding the system in a given state corresponding to the chosen basis.
The figure below shows how the wave function (blue curve) and its squared modulus (red dashed line) look. \(\psi(x)\) can change sign, positive or negative — this is the amplitude of possibility, while \(|\psi|^2\) is always positive and shows where the system is most likely to appear.
Thus, \(\psi\) describes not the event itself, but the amplitude of potentiality — that which may become real once measurement occurs.
In the position basis, \(|\psi(x)|^2\) expresses the probability of finding a particle at point \(x\); in the momentum basis — the probability of measuring a given momentum; in the energy basis — the probability of obtaining a specific energy value.
The form of the wave function changes, but its meaning remains: it describes the distribution of possibilities — the spectrum of potential answers not yet collapsed into fact.
Hence an electron around a nucleus is not a tiny sphere, but a cloud of probability — the very breath of matter, pulsating in all directions at once. On this level, matter loses its familiar outlines. Reality ceases to be solid. It becomes the respiration of possibilities — a soft pulse of probability awaiting the moment of awareness.
Superposition
In classical physics, every body has well-defined parameters: position, velocity, energy. A quantum system, however, possesses none of these precisely before measurement. It exists in a superposition of states — a linear combination of all possible values it could take.
As in the famous double-slit experiment, a particle passes through both slits simultaneously — not by trickery, but because reality itself holds the potential to be in both places at once.
Mathematically, this is simple: if a system can be in states \(\psi_1, \psi_2, \psi_3 …\), its complete state can be written as their sum:
\( \Psi = c_1\psi_1 + c_2\psi_2 + c_3\psi_3 + … \)
This ability to add states is not an accident — it follows directly from the linearity of Schrödinger’s equation. If \(\psi_1\) and \(\psi_2\) are both solutions, then any linear combination \(c_1\psi_1 + c_2\psi_2\) is also a solution. This simple mathematical fact makes possible the existence of superposition — a state in which the system simultaneously inhabits all versions of its being.
The coefficients \(c_n\) define probability amplitudes, the contribution of each variant to the total state. Until the system is measured, all these possibilities coexist — not as hidden variables, but as real components of one wave whole.
This “simultaneous being of the incompatible” is Schrödinger’s true revolution: reality ceased to be a set of facts and became an interference structure of possibilities.
Superposition is not “uncertainty” in the sense of ignorance, but active potentiality — a field of meanings where each possibility resonates with the others until an act of distinction occurs.
Decoherence — The Transition from Possibility to Fact
If superposition is the holistic cloud of possible states, then coherence means their mutual harmony — the ability of these states to interfere, to overlap like waves, strengthening or weakening one another depending on phase. Coherence allows the system to exist as a unified wave whole.
Decoherence is the process by which this harmony is lost. When a quantum system interacts with its environment (air, heat, light), the phases of its components drift apart. Interference disappears, and the system stops behaving like a wave — it begins to appear as a classical object with definite properties.
In truth, decoherence does not destroy the wave itself; it erases phase information — the delicate structure that kept the states aligned. The wave does not “collapse”; it simply loses internal coherence, turning from a living interference pattern of possibilities into a statistical distribution of facts.
From a physical standpoint, decoherence explains why we perceive the world as classical, but not why one particular classical world is realized out of the many possible ones.
Macroscopic bodies constantly exchange information with their environment and therefore cannot retain quantum coherence. A table, a stone, or a planet cannot be “in superposition,” though at a deeper level they are still governed by the same Schrödinger equation.
Philosophically, something more profound is revealed: decoherence is not merely a physical process but a description of a boundary, arising wherever consciousness draws a distinction — not by will, but by its very nature to distinguish.
Physics provides no fixed rule for where the system ends and the environment begins. But phenomenologically one can say that this boundary is established by the act of distinction — by consciousness itself deciding what counts as “inside” and what as “outside.”
The theory precisely describes how wave phases desynchronize upon environmental contact — yet the very line between “system” and “environment” is not given by nature. It is defined by the observer. For the electron, the world might be the atom; for the atom — the molecule; for the human being — the entire universe.
Consciousness establishes the frame of distinction in which this process becomes observable. It defines what counts as the system and what as background, thereby fixing the moment of transition from coherence to decoherence. In this sense, observation does not destroy the quantum wave but shapes it — transforming a spectrum of possibilities into a concrete manifestation.
Thus, decoherence can be described physically, yet its boundaries are not dictated by nature — they arise where consciousness draws the line of differentiation. It reflects not only properties of matter but also the structure of the observing act itself: reality emerges not through mechanical interaction but in the moment when consciousness separates the possible from the actual.
Perhaps the wave function is more than a mathematical construct; perhaps it mirrors a deeper principle: everything that exists manifests through a field of meanings whose probabilities become facts in the act of awareness.
Future essays will explore this theme further — coherence as both physical and semantic phenomenon, the observer’s scale as a determinant of the system–environment boundary, and why this boundary is not physical but conceptual.
Consciousness and Choice — The Act of Completing Reality
Even after decoherence, one mystery remains: why does a single outcome occur? Physics provides the tools of description but not of choice. Schrödinger’s equation is linear; it contains no mechanism for “collapse.” The step from possibility to actuality is not written in mathematics — it simply happens. And there lies the frontier between physics and metaphysics.
In Everett’s Many-Worlds interpretation, choice does not occur at all — every branch remains real, and the observer “splits” along with them. But this solution only moves the question elsewhere: why is this particular branching experienced as our world?
Here arises the hypothesis that consciousness plays a special role — not as mystical force, but as a mechanism of fixation of information. Awareness of measurement can be viewed as the completion of decoherence: not merely the loss of phase, but the anchoring of outcome within the structure of experience.
From this standpoint, consciousness is not an external spectator but a function of distinction, selecting one meaning from the manifold of possibilities. It does not create the electron nor violate physical law, yet it turns the event into a fact.
One might say that consciousness is the final stage of the wave process — the moment when potential becomes knowledge. Without it, there is no difference between the possible and the actual. And difference itself — that is the act of creation.
In that act occurs the final transition from the quantum field of probabilities to the world we call “real.”
Thus physics, which began as the study of matter, culminates as the study of meaning. The question is no longer what is, but how the possible becomes visible.
Conclusion
The Schrödinger function is the breath of probability, a map of the universe’s potential meanings. The equation \( i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi \) does not merely describe the motion of particles — it reveals that existence itself is the unfolding of possibilities.
When Schrödinger wrote his equation, he may not have known he was creating not just a formula but a mirror of the cosmos — an equation in which matter first learns to breathe with meaning.
Every measurement is an inhalation. Every realization — an exhalation. Between them, the universe lives.
The world is not given — it pulses between probability and awareness, between the breath of meaning and the form we call reality.
