Heisenberg Uncertainty Principle Explained with Examples

A Limit Built into Nature

In everyday life we assume that, with good enough instruments, we could know exactly where a car is and exactly how fast it is going at the same moment. In 1927 Werner Heisenberg showed that, at the scale of atoms and electrons, this is impossible. The more precisely you know a particle's position, the less precisely you can know its momentum, and vice versa. This is not a problem with our instruments; it is a basic property of nature. The Heisenberg uncertainty principle explains why atoms don't collapse, why electrons can't sit inside nuclei, and why spectral lines have a natural width. This tutorial states the principle, explains where it comes from, works through numerical examples, and covers the energy–time form and common misunderstandings.

Statement of the Heisenberg Uncertainty Principle

For a particle moving along the x-axis, the uncertainty in its position, Δx, and the uncertainty in its momentum, Δp, satisfy

Δx Δp ≥ ħ/2,

where ħ ("h-bar") = h/2π = 1.055 × 10⁻³⁴ J s is the reduced Planck constant. Δx and Δp are standard deviations: the spread of results you would get if you measured many identically prepared particles.

Many textbooks and exam questions use the rougher form Δx Δp ≈ h/4π (which is the same thing) or even Δx Δp ≈ h for order-of-magnitude estimates. Always check which version a question expects.

The product of the two uncertainties has a minimum value. Squeezing Δx down forces Δp up, and squeezing Δp down forces Δx up.

Where the Uncertainty Comes From

The principle follows from the wave nature of matter. A particle is described by a wave, and its de Broglie wavelength is λ = h/p.

A wave with a single, exact wavelength (and so an exact momentum) is a perfectly regular wave that goes on forever. It is spread over all space, so its position is completely uncertain.
To make a wave that is localised in a small region (a "wave packet"), you must add together many waves with a range of wavelengths, so that they reinforce in one place and cancel everywhere else. A range of wavelengths means a range of momenta.

The narrower the packet, the wider the range of wavelengths needed. This is a mathematical property of all waves; sound engineers meet the same trade-off between the duration of a sound and the spread of its frequencies.

Heisenberg's microscope. Heisenberg also gave a physical picture. To see where an electron is, you must bounce at least one photon off it. To locate it precisely you need a short wavelength, but short-wavelength photons carry large momentum (p = h/λ) and give the electron an unpredictable kick, as in Compton scattering. Using longer wavelengths disturbs it less but blurs its position.

Example 1: An Electron in an Atom

An electron is confined within an atom of diameter about 1.0 × 10⁻¹⁰ m. Estimate the minimum uncertainty in its momentum and the corresponding speed and energy.

Taking Δx ≈ 1.0 × 10⁻¹⁰ m: Δp ≥ ħ/(2Δx) = 1.055 × 10⁻³⁴ / (2 × 1.0 × 10⁻¹⁰) ≈ 5.3 × 10⁻²⁵ kg m/s.

The corresponding speed is Δp/m = 5.3 × 10⁻²⁵ / 9.11 × 10⁻³¹ ≈ 5.8 × 10⁵ m/s, and the kinetic energy p²/2m ≈ 1.5 × 10⁻¹⁹ J ≈ 1 eV.

This is the right order of magnitude for the energies of electrons in atoms (a few eV), showing that confinement alone forces electrons to move quickly. This also explains why atoms don't collapse: classically, an electron should spiral into the nucleus, but squeezing it into a tiny space would give it an enormous momentum and kinetic energy. The atom settles at the size where the total energy is lowest.

Example 2: A Cricket Ball

The position of a 0.16 kg cricket ball is known to within 1 μm (10⁻⁶ m). Find the minimum uncertainty in its velocity.

Δv ≥ ħ/(2mΔx) = 1.055 × 10⁻³⁴ / (2 × 0.16 × 10⁻⁶) ≈ 3.3 × 10⁻²⁸ m/s.

This is unimaginably small, far below anything that could ever be measured. Because ħ is so tiny, the uncertainty principle has no noticeable effect on everyday objects; it only matters for very light particles in very small spaces.

Example 3: Why Electrons Can't Live in the Nucleus

Before the neutron was discovered, some physicists thought nuclei contained electrons. Suppose an electron were confined within a nucleus, Δx ≈ 1.0 × 10⁻¹⁴ m.

Δp ≥ ħ/(2Δx) ≈ 5.3 × 10⁻²¹ kg m/s. An electron with this momentum is highly relativistic, with energy roughly pc ≈ 5.3 × 10⁻²¹ × 3.0 × 10⁸ ≈ 1.6 × 10⁻¹² J ≈ 10 MeV.

But electrons emitted in beta decay typically have energies of only about 1 MeV, and no force in the nucleus could hold an electron with 10 MeV. So electrons cannot be permanent residents of the nucleus; the electron in beta decay is created at the moment of decay.

The Energy–Time Uncertainty Relation

A similar relation links the uncertainty in a system's energy to the time available to measure it, or the lifetime of a state:

ΔE Δt ≥ ħ/2.

Example 4. An excited state of an atom has a lifetime of about 1.0 × 10⁻⁸ s. Estimate the uncertainty in its energy.

ΔE ≥ ħ/(2Δt) = 1.055 × 10⁻³⁴ / (2 × 1.0 × 10⁻⁸) ≈ 5.3 × 10⁻²⁷ J ≈ 3.3 × 10⁻⁸ eV.

So photons emitted when this state decays don't all have exactly the same energy; there is a tiny spread. This is the natural line width of spectral lines: short-lived states give broader lines. In particle physics, extremely short-lived particles have such large energy uncertainties that their masses are measured as broad peaks rather than sharp values.

Common Misunderstandings

"It's just about clumsy measurement." Heisenberg's microscope is a helpful picture, but the principle is deeper: a particle does not have an exact position and exact momentum at the same time, whether or not anyone measures it.
"Nothing can be known precisely." Position alone can be measured as precisely as you like; so can momentum alone. The limit applies to the product of uncertainties in pairs of quantities such as x and pₓ.
"It applies to x and the y-momentum." No: x and py can both be known precisely. The principle links each position coordinate with the momentum in the same direction.

Consequences

Zero-point energy: a particle confined in a box or oscillating in a potential can never be completely at rest, because Δp = 0 would need infinite Δx. Even at absolute zero, atoms keep a minimum vibration. This is why liquid helium does not freeze at normal pressure even near 0 K.
Quantum tunnelling and many features of atomic structure follow from the same wave behaviour.
Electron microscopes and particle accelerators use very high-momentum (short-wavelength) particles to probe small distances, the same trade-off seen in Heisenberg's microscope.

Summary

Heisenberg's uncertainty principle: Δx Δp ≥ ħ/2, with ħ = 1.055 × 10⁻³⁴ J s.
It comes from the wave nature of particles: a localised wave packet needs a spread of wavelengths, and so of momenta.
For electrons in atoms, confinement gives energies of a few eV; for everyday objects the effect is negligible.
Energy–time form: ΔE Δt ≥ ħ/2, giving the natural width of spectral lines.
It explains atomic stability, zero-point energy and why electrons cannot be confined in nuclei.