The Key Role of Quantum Uncertainty: No Particle Can Be at Rest

Einsenberg's Formula

Einsenberg's formula is a fundamental discovery of quantum mechanics. Formulated in 1927 by Werner Einsenberg (1901-1976), it expresses a fundamental limit to the precision with which certain pairs of physical properties, such as position (x) and momentum (p), can be simultaneously known.

This relationship is mathematically expressed by the inequality: Δx * Δp ≥ ħ/2, where \(\hbar = \frac{h}{2\pi}\) (h = Planck's constant and ħ = reduced Planck's constant). The inequality (≥) means that the product of the two uncertainties (Δx) and (Δp) can never fall below ħ/2.

In other words: The more precisely the position of a particle (Δx) is measured, the less precisely its momentum (Δp) can be known, and vice versa. This is a fundamental limit of nature, not a technical limitation of our instruments. We can never know both the position and the velocity of a quantum particle simultaneously with absolute precision.

Interpretation of the terms Δx * Δp ≥ ħ/2

Left side: Δx * Δp

Δx (delta x) represents the uncertainty in the position of a particle. It is the "margin of error" or imprecision with which we know exactly where the particle is in space. Δp (delta p) represents the uncertainty in the momentum (or impulse) of the particle. The momentum p = mv (mass × velocity), so Δp measures the imprecision in the speed/movement of the particle. The product Δx * Δp therefore represents the product of these two uncertainties.

Right side: ħ/2

ħ (h-bar) is the reduced Planck constant, equal to h/2π, where h is Planck's constant (approximately 1.054 × 10⁻³⁴ J·s). It is a fundamental constant of nature that characterizes the quantum scale. ħ/2 is therefore half of this constant.

A Limit Imperceptible at Our Scale

The smallness of ħ does not mean that at our macroscopic scale, we "know" the position and movement of an object. It means that our usual measurement inaccuracies are enormous compared to ħ/2, thus masking the quantum effects of macroscopic objects. The inequality is always widely verified in practice.

At the quantum scale, our measurement precisions are on the order of 10⁻¹⁵ m, far from ħ/2. What changes is that the compromise imposed by ħ/2 becomes relevant for understanding the behavior of particles. The uncertainty principle becomes physically significant at this scale because confining an electron in a space of 10⁻¹⁰ m (the size of an atom) automatically implies that it has movement, and reducing Δx requires increasing Δp according to the relation ΔxΔp ≥ ħ/2.
An electron that is both immobile and localized is therefore impossible, which explains why no particle can be at rest.

Example to Visualize the Idea of Heisenberg's Uncertainty Principle

Imagine you are trying to take a very sharp photo of a car moving at high speed. To get a sharp image, you must use a very short exposure time. However, a short exposure time means you capture less light, which can make the image dark or blurry if the light is insufficient.

• Precise position (sharp photo): If you manage to get a very sharp photo of the car, you have captured its position at a precise moment. However, due to the short exposure time, you have less information about its movement (speed), as you only see a snapshot without context of displacement.
• Precise movement (blurry photo): If you use a longer exposure time to capture more light and get a brighter image, the car will appear blurry due to its movement during the exposure. You then have more information about the movement (you see the blur indicating direction and speed), but you lose precision on the exact position of the car at a given moment.

Impacts on Modern Physics

• Reality at the quantum scale: The uncertainty principle has allowed the exact prediction of the evolution of systems. This principle shows that reality at the quantum scale is intrinsically probabilistic. It has replaced the deterministic view of classical physics with a probabilistic framework.
• Wave function collapse: Uncertainty implies that measuring a quantity can disturb the system, making the notion of an "objective observer" problematic. The notion of "wave function collapse," which postulates that a measurement causes the wave function of a system to "collapse" into a defined state, emerged in part as a consequence of the uncertainty principle.
• Energy quantization: The uncertainty principle is crucial for explaining phenomena such as energy quantization in systems like atoms (Bohr model) and molecules. Quantum effects influence macroscopic phenomena such as superconductivity and superfluidity.
• Tunnel effect: The uncertainty principle is also fundamental for explaining the tunnel effect, a quantum phenomenon where a particle has a non-zero probability of passing through an energy barrier even if it does not have enough energy to overcome it according to the laws of classical physics. This phenomenon is used in devices such as tunnel effect transistors and tunnel effect microscopes.
• Quantum vacuum: Due to uncertainty, even a quantum vacuum (the ground state of a system) is not completely "empty" but is subject to quantum fluctuations.
• Wave-particle duality: Uncertainty is intimately linked to wave-particle duality, which describes how quantum objects (such as electrons or photons) can behave as both waves and particles. The uncertainty in position and momentum is a direct manifestation of this duality.