Does the electron have a shape? The surprising answer of quantum physics

The point-like electron of the Standard Model

A particle with no spatial extension

In the Standard Model, the electron is described as a point particle. This means it has no intrinsic size: its charge and mass are concentrated at a mathematical point. This description is extraordinarily effective: it allows calculating with spectacular precision quantities such as the magnetic moment of the electron, whose value predicted by quantum electrodynamics (QED) coincides with experiment to more than ten decimal places.

But be careful: "point-like" does not mean "without structure." In QED, the electron is surrounded by a cloud of virtual particles (photons, electron-positron pairs) that screen its charge. This screening structure is responsible for measurable effects, such as the Lamb shift or the anomalous magnetic moment. The "shape" of the electron, in this framework, is therefore that of a point surrounded by a cloud of virtual particles from quantum vacuum fluctuations.

Experimental limits on the size of the electron

Physicists have attempted to measure the size of the electron by studying its scattering on other particles. If the electron had a spatial extension, the scattering results would deviate from point-like predictions. To date, no deviation has been observed. The strictest constraints give an upper limit of the order of \(10^{-18}\) meters, or one thousandth of the size of a proton. The electron is therefore, to our knowledge, truly point-like.

The wave function: the probabilistic shape of the electron

A haze of probability

If the electron is point-like, how can it have a "shape"? The answer lies in the wave function \(\psi(r,t)\), introduced by Erwin Schrödinger (1887-1961). This function does not describe a physical material wave, but a probability amplitude. Its square \(|\psi|^2\) gives the probability density of finding the electron at a given position during a measurement.

It is this probability distribution that constitutes the "shape" of the electron in quantum mechanics. Around an atomic nucleus, for example, the electron forms an electron cloud whose geometry depends on its quantum state. This shape is not fixed: it evolves according to the Schrödinger equation and deforms under the effect of external fields.

Wave function collapse

When the electron's position is measured, the wave function collapses to a precise point: this is wave function collapse. Before measurement, the electron has no defined position; its "shape" is a cloud of possibilities. After measurement, it appears locally as a particle. This duality between extended shape and point-like appearance is at the heart of the Copenhagen interpretation, defended by Niels Bohr (1885-1962).

Atomic orbitals: the shape of the electron in the atom

Well-defined three-dimensional patterns

In an atom, electrons occupy quantum states called atomic orbitals. Each orbital is characterized by three quantum numbers (n, l, m) and has a precise geometric shape. The s orbital (l = 0) is spherical. The p orbitals (l = 1) have a dumbbell shape, with two lobes oriented along the x, y or z axes. The d orbitals (l = 2) have four-lobed shapes, and so on.

These shapes are not mere mathematical curiosities: they determine the geometry of chemical bonds and therefore the structure of molecules. The shape of the electron in an atom is thus directly responsible for the shape of matter at our scale.

Table of orbitals

The table below summarizes the main shapes of atomic orbitals and their characteristics.

The sphericity of the electron: what do experiments say?

An almost perfect charge distribution

If the electron is point-like, how can we speak of its "shape"? In reality, even a point particle can have an effective charge distribution due to quantum fluctuations. Physicists seek to know whether this distribution is perfectly spherical or slightly deformed. A deformation would reveal the existence of an electric dipole moment (eEDM), a property that would violate certain fundamental symmetries.

The most sensitive experiments use polar molecules (such as thorium fluoride, ThF) trapped in intense electric fields. If the electron had an eEDM, its spin would align slightly with the electric field, producing a measurable shift. To date, no signal has been detected, which imposes an extremely strict upper limit.

Current limits on the eEDM

In 2018, the ACME collaboration established a limit of \(|d_e| < 1.1 \times 10^{-29} \, e\cdot\text{cm}\). In 2023, a new measurement further improved this constraint. Future experiments aim for a sensitivity of \(10^{-30} \, e\cdot\text{cm}\), which would probe energies well beyond those accessible at the LHC. 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, and could explain the matter-antimatter asymmetry of the Universe.

The form factor: how do we measure the shape of the electron?

Electron scattering

To probe the shape of a particle, physicists use scattering. By projecting electrons onto a target (protons, nuclei, other electrons), one measures how they are deflected. If the target particle is point-like, the cross section follows a simple law (Rutherford formula). If it has a spatial extension, the scattering is modulated by a form factor \(F(q^2)\), where \(q\) is the momentum transfer.

Applied to the electron itself, this reasoning gives a limit on its size. Electron-electron and electron-positron scattering experiments at LEP and SLAC have shown that the electron behaves as a point particle down to scales of \(10^{-18}\) meters. No non-trivial form factor has been observed.

Radiative corrections

Even point-like, the electron acquires an effective structure through its interactions with the quantum vacuum. The radiative corrections of QED modify its charge and magnetic moment. The magnetic moment of the electron is given by \(g/2 = 1 + a_e\), where \(a_e\) is the magnetic anomaly. The measured value of \(a_e\) is one of the most precise in all of physics: \(a_e = 0.001\,159\,652\,180\,73(28)\). This agreement between theory and experiment is a spectacular proof of the validity of QED, but it reveals no deformation of shape.

Synthesis: the multiple faces of the electron's shape

The table below summarizes the different ways of describing the "shape" of the electron according to the theoretical framework and the experimental viewpoint.

New frontiers: the shape of 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 particle-like behavior. This experiment confirms that the "shape" of the electron depends on the measurement device: it is not an intrinsic property, but a relational one.

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. The "shape" of the electron there appears as a diffracted matter wave.

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 decoherence," shows that decoherence can occur at a distance, with consequences for electron microscopy and interferometers. The quantum shape of the electron gradually degrades under the effect of the environment.