The Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle states that the position and the momentum of a particle cannot both be known exactly at the same time: Δx · Δp ≥ ħ/2, where ħ = h/2π = 1.055 × 10−34 J·s. In brief: if an electron is confined to a region the size of an atom (Δx = 1.0 × 10−10 m), its velocity must be uncertain by at least 5.8 × 105 m/s, whereas for a baseball the limit is far too small to matter.

The uncertainty principle is not a limitation of our instruments but a fundamental property of nature, which follows from the wave character of matter. It explains why electrons do not fall into the nucleus, why atoms have a certain size and why a particle in a box has a minimum energy. This article states the principle, gives calculations for objects of different sizes in a table, explains its link to zero-point energy and describes common misunderstandings.

What does the principle say?

In one dimension, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp = mΔv) is at least a certain minimum:

Δx · Δp ≥ ħ / 2, where ħ = h / 2π = 1.0546 × 10−34 J·s

Some textbooks use the simpler form Δx · Δp ≥ h/4π, which is the same, or the approximate form Δx · Δp ≈ h. The more precisely the position is known (small Δx), the less precisely the momentum is known, and vice versa. Position and momentum are an example of a pair of “conjugate” quantities. The uncertainties are the standard deviations of many measurements on identically prepared systems, not the errors of one measurement.

Why does it arise?

A particle is described by a wave. A wave with a single wavelength extends through all space, so it has a precisely known momentum (p = h/λ) but no definite position. To localize the wave to a small region, many waves with different wavelengths must be added, and then the momentum is not definite. This is a general property of waves: a short pulse always contains a wide range of frequencies. It is not caused by the disturbance of the system by the measurement, although a measurement with a photon does disturb an electron; the principle holds even in the absence of measurement.

Δx x p Δp Δx x p Δp

Uncertainty for different objects

Minimum uncertainty in speed, Δv ≥ ħ/(2mΔx)
Object Mass (kg) Δx Minimum Δv (m/s)
Electron in an atom9.11 × 10−311.0 × 10−10 m5.8 × 105
Electron in a nucleus-sized region9.11 × 10−311.0 × 10−14 m5.8 × 109
Dust grain (1 µg)1.0 × 10−91.0 × 10−6 m5.3 × 10−20
Baseball0.1451.0 × 10−9 m3.6 × 10−25

The electron in an atom has an unavoidable uncertainty in speed that is a sizable fraction of the speed of light (0.2 %, here); for a nucleus-sized region, the uncertainty is larger than the speed of light, so an electron cannot be confined in a nucleus (the relation then needs a relativistic treatment, but the conclusion is the same). For macroscopic objects the principle has no practical effect.

Worked examples

Example 1: an electron in an atom

Δx = 1.0 × 10−10 m. Δp ≥ ħ / (2Δx) = 1.0546 × 10−34 / (2 × 1.0 × 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.

Example 2: a baseball

For a baseball of mass 0.145 kg localized to 1 nm, Δv ≥ 1.0546 × 10−34 / (2 × 0.145 × 1.0 × 10−9) = 3.6 × 10−25 m/s. The baseball moves 10−25 m in a second because of this uncertainty, which is a distance far smaller than a nucleus.

Example 3: zero-point energy of a particle in a box

An electron is confined to a one-dimensional box of length L = 1.0 nm. The uncertainty in position is of the order of L, and therefore, by the principle, the momentum cannot be zero. The minimum energy (n = 1) from the exact solution is E1 = h2 / (8mL2) = (6.626 × 10−34)2 / (8 × 9.109 × 10−31 × (1.0 × 10−9)2) = 4.39 × 10−67 / 7.29 × 10−48 = 6.0 × 10−20 J = 0.38 eV. This is the zero-point energy. An estimate from Δp = ħ/(2Δx) with Δx = L gives a kinetic energy of the same order, p2/2m ≈ 1.5 × 10−21 J (an underestimate, since Δp is only a lower limit for the momentum).

Example 4: speed of a localized proton

A proton (1.673 × 10−27 kg) is localized to within the diameter of a nucleus, Δx = 1.0 × 10−14 m. Δv ≥ 1.0546 × 10−34 / (2 × 1.673 × 10−27 × 1.0 × 10−14) = 3.2 × 106 m/s, about 1 % of the speed of light, which agrees with the large kinetic energies of nucleons in nuclei.

What is the energy–time form?

A similar relation holds for energy and time: ΔE · Δt ≥ ħ/2. A state that lives for a short time Δt has an energy that is not sharply defined. This explains the natural width of spectral lines: excited states with a short lifetime (about 10−8 s for typical atomic states) give lines with a width of ΔE ≈ ħ/Δt ≈ 10−26 J, or a frequency spread of about 107 Hz. States with a very long lifetime give very sharp lines, which is the basis of atomic clocks.

What are the consequences for chemistry?

Who formulated the principle?

Werner Heisenberg published the principle in 1927, when he was 25 years old, in the context of his formulation of matrix mechanics. He illustrated it with the thought experiment of a “gamma-ray microscope”. Earle Kennard and Hermann Weyl derived the exact inequality with ħ/2 soon after. Heisenberg received the Nobel Prize in Physics in 1932. Niels Bohr developed the related concept of complementarity at the same time.

Which mistakes are common?