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Planck units

Planck units are a system of units of measurement based on the fundamental constants:

The Planck units are often semi-humorously referred to by physicists as "God's units". They eliminate all arbitrariness from the system of units: an extra-terrestrial intelligence can be expected to use the same system. These units have the advantage of simplifying many equations, because expressed in Planck units, all the constants on which they are based have the numerical value 1. For this reason, the units are popular in quantum gravity research. However, they are too small for practical use, unless prefixed with large powers of ten. They also suffer from uncertainties in the measurement of some of the constants on which they are based, especially of the gravitational constant G.

The Planck units are:

Length: (h G / c³)^1/2 ≈ 1.6 × 10^-35 meters (the Planck length)
Mass: (c h / G)^1/2 ≈ 2.1 × 10^-8 kilograms (the Planck mass)
Time: (h G / c⁵)^1/2≈ 5.4 × 10^-44 seconds (the Planck time)
Temperature: (c⁵ h / G)^1/2 / k ≈ 1.4 × 10³² kelvin (the Planck temperature)
Density: (Planck Mass / (Planck Length)³) ≈ 5.1 × 10⁹⁶[?] kilograms/meters³ (the Planck density)

At the "Planck scales" in length, time, or temperature the effects of quantum physics dominate the behavior of the system. The Planck Density is extremely high, above which our ideas of gravity and quantum mechanics break down.

The NIST (National Institute of Standards and Technology) website is a convenient source of data on the commonly-recognized constants, including Planck units.

Max Planck's creation of the natural units

Max Planck first listed his set of units (and gave values for them remarkably close to those used today) in May of 1899 in a paper presented to the Prussian Academy of Sciences. Max Planck: 'Über irreversible Strahlungsvorgänge'. Sitzungsberichte der Preußischen Akademie der Wissenschaften, vol. 5, p. 479 (1899). At the time he presented the units, quantum mechanics had not been invented. He himself had not yet discovered the theory of blackbody radiation (first published December 1900) in which the constant h made its first appearance and for which Planck was later awarded the Nobel prize. The relevant parts of Planck's 1899 paper leave some confusion as to how he managed to come up with the units of time, length, mass, temperature etc. which today we define using h-bar and motivate by references to quantum physics before things like h-bar and quantum physics were known. Here's a quote from the 1899 paper that gives an idea of how Planck thought about the set of units.

...ihre Bedeutung für alle Zeiten und für alle, auch ausserirdische und ausser menschliche Culturen nothwendig behalten und welche daher als 'natürliche Maasseinheiten' bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as 'natural units'...