Heisenberg Uncertainty Principle
Position and momentum cannot both be known with unlimited precision.
Formula
Variables
Example
Pin down an electron’s position precisely and its momentum becomes hopelessly uncertain.
How it works
In quantum mechanics, a particle is not described as having an exact point-position and exact momentum at the same time; instead, its state is a wavefunction spread across space and momentum. The inequality Delta x Delta p hbar/2 follows from the wave-like nature of matter and the mathematical fact that a sharply localized wave packet must be built from many wavelengths, which means a wide spread of momenta. So the more tightly a state is confined in space, the less narrowly its momentum can be defined.
When to use it
Use this when analyzing quantum particles such as electrons, atoms, or photons, especially in confinement problems, wave packets, and measurement limits. It is also essential in explaining why bound particles in atoms and nanoscale devices cannot have simultaneously exact position and momentum.
Worked example, step by step
Suppose an electron is confined to a region with position uncertainty Delta x = 2.0 10^-10 m. Use Delta x Delta p hbar/2, so Delta p (hbar)/(2Delta x). Take hbar = 1.055 10^-34 J s. Delta p (1.055 10^-34) / (2 2.0 10^-10) Delta p (1.055 10^-34) / (4.0 10^-10) Delta p 2.64 10^-25 kg m/s Final answer: the electron's momentum uncertainty must be at least about 2.6 10^-25 kg m/s.
Common mistakes to avoid
- Treating the relation as a limitation of measurement instruments only, instead of a property of quantum states
- Using hbar where the problem actually gives Planck's constant h, or forgetting the factor of 1/2
- Assuming Delta x and Delta p are exact errors from a single measurement rather than statistical spreads in many identically prepared systems
Did You Know?
You cannot know a particle's exact position and momentum at once — nature has a built-in fuzziness.
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