Planck length

The Planck length is the natural unit of length, denoted by <math>\ell_P<math>.

History

This unit was first developed by Max Planck who wished to create a system of measurement based on natural units. These are all based on the Planck mass. Although quantum mechanics and general relativity were unknown at the time that the units were proposed, it later became clear that at distances of the Planck length, gravity would begin to display quantum mechanical effects, requiring a theory of quantum gravity to predict what happens.

Value

The Planck mass is a mass whose Schwarzschild radius and its Compton length are equal distances. This distance, called the Planck length, is equal to:

<math> \ell_P =(\hbar G/c^3)^{1/2} \cong 1.616 \times 10^{-35}<math> metres

where:

<math>\hbar<math> is Dirac's constant

G is the gravitational constant

c is the speed of light in vacuum

The estimated size of the Universe (7.4 × 10²⁶ m) is 1.2 × 10⁶² Planck lengths.

Consequences

By the Heisenberg uncertainty principle of standard quantum mechanics, an object whose position was accurate to the Planck length would have an uncertainty in momentum approximately 3.2629 kg m / s. What this means is, if one could use some hypothetical apparatus to determine the position of a baseball (or any other object of the same mass) and be accurate to the Planck length at a given moment, it would be impossible to distinguish a speed of zero (at rest) from a speed of 22.89 m/s (approximately 51 miles an hour).