From the probability cloud to the particle: the electron according to quantum mechanics

The probability wave defies our intuition of the world!

A haze of probability rather than a localized ball

Imagine an electron as a light mist spread through space rather than a well-localized ball in time. Even with the best microscopes or the most powerful calculations, you will never be able to point at it and say: "It is there!" The great quantum mystery is that at the moment of measurement, it "chooses" an exact position, as if all the mist suddenly condensed into a drop of water.

The wave function and probability density

This strange dance between blurriness and precision is at the heart of the quantum revolution. In quantum mechanics, the electron (an elementary particle of negative charge) is not described as a corpuscle, but as a wave function \(\psi(r,t)\) introduced by Erwin Schrödinger (1887-1961). This function encodes the probability density \(|\psi|^2\), which gives the chance of finding the electron at a given position. The image of a probability cloud therefore replaces the classical idea of a well-defined trajectory.

The Copenhagen interpretation and the reduction of the wave packet

The wave function is not an extended physical reality but a probabilistic prediction tool, as Niels Bohr (1885-1962) said in the Copenhagen school. However, during a measurement, for example on a fluorescent screen, the electron is detected at a precise point, like a particle. This phenomenon is called reduction of the wave packet (collapse of the wave function).

The fundamental limits of quantum knowledge

Heisenberg's uncertainty principle forbids knowing simultaneously its position and its momentum with arbitrary precision. This strangeness is not specific to electrons: interference experiments carried out on increasingly massive molecules show that quantum behavior extends well beyond the atom. What limits its observation is not a theoretical boundary, but decoherence, which erases quantum properties as soon as a system interacts too strongly with its environment.

Thus, the electron truly exists, but as a relational quantum entity: its intrinsic properties (charge, mass, spin) are invariant and define its identity, while its positional and dynamical properties depend on its interaction with the environment. The electron is not an isolated object; it is a node of quantum relations within a universe in permanent interaction.

The electron in an electrical wire: wave or ball?

Neither a ball nor a classical wave

In an electrical wire, the conduction electron is not a small ball moving along the cable. The charged-ball model, called the Drude model, is a useful approximation for explaining Ohm's law and resistance, but it does not correctly describe quantum reality. The electron is also not a classical wave, like a wave oscillating in a material medium.

A delocalized quantum object

The electron is a quantum object described by a wave function. In a metal, conduction electrons occupy extended states called Bloch waves. They are delocalized over the entire crystal and form a Fermi sea. One therefore cannot follow an individual electron like a ball, nor assign it a precise trajectory without perturbing the system.

Current is not a flow of balls

In a wire, electric current does not correspond to electrons rushing at the speed of light. The disordered velocity of electrons, linked to the Fermi velocity, is very large, on the order of 106 m/s. On the other hand, the collective drift velocity, the one that produces current, is much lower, often a few fractions of a millimeter per second. What propagates rapidly is the electromagnetic field that sets the whole set of electrons in motion.

An appearance of particle during measurement

When an electron interacts with a detector or an impurity, it manifests locally as a particle. But as long as it remains in the wire, it is better described as a delocalized quantum wave. Thus, the electron is neither a ball nor a classical wave: it is a quantum object that presents wave-like or particle-like aspects depending on the context.

The elementary charge: a fundamental constant of physics

The electron charge is a fundamental constant of physics, denoted −e (or −1.602176634 × 10⁻¹⁹ Coulomb in absolute value). It represents the smallest amount of electric charge freely observable in nature. This value, measured with extreme precision, is the basis of quantum electrodynamics and defines the unit of charge in the International System (SI) since the 2019 redefinition, where the coulomb is now linked to the elementary charge via the Planck constant. Practical equivalence: 1 C ≈ charge of \(6{,}24 \times 10^{18}\) electrons.

Classical model vs. quantum description

Heisenberg's uncertainty principle

Unlike Newtonian mechanics, where position and velocity are defined simultaneously, the uncertainty principle of Werner Heisenberg (1901-1976) states that \(\Delta x \cdot \Delta p \geq \hbar / 2\). This implies that the better one knows the electron's position (x), the less one knows its momentum (p), and vice versa.

The table below summarizes the essential differences between the classical description of a particle and its quantum description.

The quantum boundary: how far does the strangeness of particles apply?

Quantum mechanics is not limited to electrons! Its rules apply to all fundamental particles (photons, quarks, neutrinos) but also to objects much larger than one imagines. Yet a question persists: where does the reign of the quantum stop? The answer is surprising: there is no theoretical limit… only technical challenges to observing these effects on a large scale.

From electrons to molecules: the quantum at work

Experiments confirm that even complex molecules (such as buckminsterfullerene, a molecule of 60 carbon atoms) can show wave-like behaviors. In 2019, physicists observed interference with molecules of 2,000 atoms! These objects, although massive, obey quantum laws as long as they remain isolated from their environment (without collisions, heat or intrusive measurement).

Experiments with giant molecules use high-vacuum chambers to avoid any parasitic interaction. Experiments with organic molecules work at room temperature if the observation time is ultra-short (nanoseconds). Experiments with macroscopic systems, such as the superconducting circuits of quantum computers, use cryogenics (−273.15°C or 0 Kelvin).

Decoherence: why are humans not quantum?

So why do we not see humans in "superposition of states"? Because of decoherence: as soon as a quantum system interacts with its environment (air, light, heat), its "magical" properties (superposition, entanglement) disappear almost instantly.

The following table illustrates the rapid drop in quantum coherence time as the size of the system increases.

N.B.:
Quantum coherence time drops strongly with the size and temperature of a system. In other words, the larger an object is and the more it is agitated by heat, the faster it switches to classical behavior.

The electron's electric dipole moment: a window onto new physics

A property beyond the Standard Model

The electron has a magnetic moment predicted with extreme precision by quantum electrodynamics. But it could also have an electric dipole moment (eEDM), an infinitesimal charge separation aligned with its spin. The Standard Model predicts it to be almost zero. If it were detected, it would violate time-reversal symmetry (T) and CP symmetry, and could explain the matter-antimatter asymmetry of the Universe.

A violation of T already observed

The violation of T has been observed in B mesons: in 2012, the BaBar experiment showed that the transformations \(B^0 \to \bar{B}^0\) and \(\bar{B}^0 \to B^0\) do not have the same probability, with a certainty of about \(14\,\sigma\). It is the same type of mechanism that could give the electron a nonzero eEDM.

Experimental limits

In 2018, the ACME collaboration established \(|d_e| < 1{,}1 \times 10^{-29} \, e\cdot\text{cm}\). In 2023, a new measurement further improved this limit. Future experiments aim for \(10^{-30} \, e\cdot\text{cm}\). The electron thus becomes a miniature laboratory for testing the limits of the Standard Model.

N.B.: If the eEDM exists, it is aligned with the electron's spin. Its discovery would require a new source of CP violation, beyond the Standard Model.

New frontiers: the electron tested by recent experiments

The “recoil slit” experiment and complementarity

In 2025, a Chinese team carried out for the first time the “recoil slit” thought experiment imagined by Einstein and Bohr in 1927. By trapping a single atom, it observed the continuous transition between wave-like and corpuscular behavior, confirming that the “wave” or “particle” character depends on the measurement device.

Positronium diffraction: a new confirmation of duality

In 2026, an international team observed for the first time the diffraction of a positronium beam, an exotic atom formed by an electron and a positron. This experiment extends wave-particle duality to a bound system and opens the way to tests of quantum gravity with antiatoms.

Radiative decoherence of free electrons

A 2024 study showed that free electrons can lose their quantum coherence over long distances simply by interacting with vacuum radiation. This effect, called “radiative coherence loss,” shows that decoherence can occur remotely, with consequences for electron microscopy and interferometers.