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In 1927, the German physicist Werner Heisenberg showed that there’s a fundamental limit to how precisely certain pairs of quantities can be known at the same time. The more precisely you know where an electron is, the less precisely you can know how fast it’s moving, and vice versa. This uncertainty principle isn’t about clumsy equipment; it’s built into nature. For chemists, it explains why electrons don’t follow neat orbits, why atoms don’t collapse, and why we describe electrons with orbitals and probabilities.
The principle
For position (x) and momentum (p = mv) along the same direction:
Δx × Δp ≥ h ÷ 4π
- Δx = uncertainty in position
- Δp = uncertainty in momentum
- h = Planck’s constant (6.626 × 10⁻³⁴ J s), so h/4π ≈ 5.27 × 10⁻³⁵ J s
(You’ll also see ħ/2, where ħ = h/2π. It’s the same thing.)
The product of the two uncertainties can never be smaller than about 5 × 10⁻³⁵ J s. If Δx is made very small, Δp must become large, and the reverse.
Where it comes from: waves
The principle follows from wave-particle duality. A particle’s momentum is linked to its wavelength by de Broglie’s relation, p = h/λ. See the de Broglie wavelength.
- A wave with a single, precise wavelength (precise momentum) is spread out endlessly, so its position is completely uncertain.
- To make a wave localised in one small region (precise position), you must combine many waves with different wavelengths, so the momentum becomes uncertain.
You can’t have a wave that is both perfectly localised and has one exact wavelength. Since electrons behave as waves, they inherit this limit.
Worked example 1: an electron in an atom
An electron is confined within an atom of diameter about 1 × 10⁻¹⁰ m. Estimate the minimum uncertainty in its velocity.
Take Δx ≈ 1 × 10⁻¹⁰ m.
Δp ≥ h/(4π Δx) = 5.27 × 10⁻³⁵ ÷ 1 × 10⁻¹⁰ = 5.27 × 10⁻²⁵ kg m/s
Δv = Δp ÷ m = 5.27 × 10⁻²⁵ ÷ 9.109 × 10⁻³¹ ≈ 5.8 × 10⁵ m/s
The uncertainty in the electron’s speed is over half a million metres per second, comparable to the speeds themselves. It’s meaningless to talk about the electron having a precise speed and position, or following a definite path around the nucleus.
Worked example 2: a football
A 0.43 kg football’s position is known to within 1 mm. Estimate the minimum uncertainty in its velocity.
Δv ≥ h/(4π m Δx) = 5.27 × 10⁻³⁵ ÷ (0.43 × 1 × 10⁻³) ≈ 1.2 × 10⁻³¹ m/s
Utterly negligible. For everyday objects, the uncertainty principle has no noticeable effect, which is why classical physics works perfectly well for footballs and planets.
The difference comes from mass: electrons are about 10³⁰ times lighter than a football.
Why electrons can’t have Bohr orbits
The Bohr model pictured electrons moving in circular orbits of exact radius at exact speeds. That would mean knowing both position and momentum precisely at every moment, which the uncertainty principle forbids.
Instead, quantum mechanics describes electrons with orbitals: regions where there’s a certain probability of finding the electron. We can’t say where the electron is at a given moment, only where it’s likely to be. See electron clouds and probability and the shapes of s, p, d and f orbitals.
Why atoms don’t collapse
Classically, the negative electron should be pulled into the positive nucleus. The uncertainty principle prevents this:
- If the electron were squeezed into the tiny nucleus (Δx ≈ 10⁻¹⁵ m), its momentum uncertainty, and therefore its typical kinetic energy, would become enormous.
- As the electron moves closer to the nucleus, its potential energy falls, but its kinetic energy (from confinement) rises.
- The atom settles at a size where the total energy is lowest: a balance between these two effects.
A rough calculation using this balance gives a hydrogen atom radius close to the actual Bohr radius, 53 pm. The uncertainty principle effectively sets the size of atoms, and therefore the size of everything made of them.
Zero-point energy
A related consequence: a particle confined in a region can never have exactly zero kinetic energy, because that would mean exactly zero momentum (Δp = 0), which is impossible if Δx is finite. The minimum energy is called the zero-point energy.
Chemical examples:
- Molecules always vibrate, even at absolute zero. Bonds are never perfectly still.
- Helium stays liquid at normal pressure even at temperatures approaching absolute zero, because its atoms are so light that their zero-point motion prevents them from settling into a solid.
- Zero-point energy differences between isotopes (for example, C–H vs C–D bonds) cause kinetic isotope effects, which chemists use to study reaction mechanisms.
Energy and time
A similar relation links energy and time:
ΔE × Δt ≥ h ÷ 4π
An excited state that exists only for a short time Δt has an uncertain energy ΔE. Consequences:
- Spectral lines have natural widths. Excited states with short lifetimes give broader lines, because the energy of the emitted photon isn’t perfectly sharp. See emission vs absorption spectra.
- In NMR spectroscopy, fast relaxation processes broaden signals. See NMR explained.
Common misconceptions
“It’s just about disturbing the particle when we measure it.” Measurement does disturb small particles (to see an electron, you must bounce a photon off it, which changes its momentum). But the uncertainty principle goes deeper: even before measurement, the particle simply doesn’t have a precise position and momentum at the same time.
“It means science can’t be precise.” Quantum mechanics makes some of the most precise predictions in all of science. The uncertainty principle limits pairs of quantities, not measurement in general. You can measure an electron’s position extremely precisely; you just can’t also know its momentum precisely at that moment.
“It applies to everything equally.” In principle, yes; in practice, it only matters for very small masses (electrons, protons, atoms). For anything you can see, the uncertainties are far too small to measure.
“It means anything is possible.” No. It’s a precise mathematical limit, and it leads to very definite predictions, such as the exact shapes and energies of orbitals.
A short history
Heisenberg developed the principle in 1927, shortly after helping to create quantum mechanics. It sparked famous debates between Albert Einstein, who disliked the idea that nature is fundamentally probabilistic (“God does not play dice”), and Niels Bohr, who defended it. Experiments ever since have supported the quantum picture. Heisenberg received the Nobel Prize in Physics in 1932 for the creation of quantum mechanics.
Key takeaways
- The uncertainty principle: Δx × Δp ≥ h/4π; position and momentum can’t both be known precisely.
- It follows from the wave nature of matter, not from imperfect equipment.
- For electrons in atoms, the uncertainty in speed is enormous, so definite orbits are impossible; orbitals describe probabilities instead.
- It explains why atoms have a stable size and why molecules have zero-point energy.
- The energy–time relation explains the natural width of spectral lines.
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