Natural units are based on universal physical constants. The units are defined so that the relevant physical constant becomes one (unity, that is). The downside is that potentially there is loss of understanding when the constants become irrelevant to statement of physical laws. Some claim this whole topic is a mind game; other say it is the only sensible way to define units. I don’t think the jury has even been chosen.
An obvious example is c, the speed of light. Picking units so that c=1 turns E=Mc2 into E=m. It means that walking speed is around 5x10-11 and the speed of sound is around 10-8.
There are several sets of units. In particle Physics, wikipedia tells me, the phrase "natural units" generally means: h=c=kB=1where h is the reduced Planck constant, c is the speed of light, and kB is the Boltzmann constant.
Planck units are similar but we add the gravitational constant G=1. Length, Lp is then such that L2=hG/c3, so one planck length is 1.616x10-35m, or a metre is around 6.168x1034 . A planck mass, mp , is mp2=hc/G, so one is 2.176x10-8kg and 1kg is 4.596x107. Planck time, tp2=hG/c5, so 1=5.39x10-44, which means I’m around 3.5x1052 in age. temperature Tp2=hc5/GK2, so 1 is 1.417x1032 K, which makes room temperature a bit low, at around 2x10-30.
I have a regular problem with units of charge, since I don’t understand what the fine-structure constant, α, might be. Besides being 0.007297.
Stoney units set G=c=kB=e=1, where e is the elementary charge and h=1/α. Stoney (wikipedia again) didn’t propose the Planck constant in 1874, since the Planck constant was not discovered (is that the right word?) until 1900.
Atomic (Hartree) units say e=me=h=KB=1 and c=1/α. In these units, the speed of light is 137, G is 10-45. I’m 7.7x1034 mass units and 7.825x1025 time units.
So these things look like they have sensible application in large scale and small scale physics, but not a lot of use in ‘real life’.
Planck constant (reduced): the quantum of angular momentum in quantum mechanics. E=hv connects the energy of a photon and the frequency of its elctromagnetic wave. Using λv=c, we could write Eλ=hc. In traditional units, h=6.626 x10-34 Joule seconds and the reduced on is that value divided by 2π, or 1.054x10-34 Js. Planck’s constant is also called the quantum of action, which (action) is something I want to chase down and attempt to understand.
Boltzmann constant (not understood by DJS) Wikipedia says: Boltzmann's constant, k, is a bridge between macroscopic and microscopic physics. Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure p and volume V is proportional to the product of amount of substance n (in moles) and absolute temperature T: p V = n R T
where R is the gas constant (8.3144621(75) J⋅K−1⋅mol−1[1]). Introducing the Boltzmann constant transforms the ideal gas law into an alternative form: p V = N k T ,
where N is the number of molecules of gas. For n = 1 mol, N is equal to the number of particles in one mole (Avogadro's number).
Boltzmann constant = 1.38064852 × 10-23 m2 kg s-2 K-1 [which is J/K. I assume K refers to Kelvin and I note a proposal to change the definition so that Boltzmann’s constant becomes exactly 1.38065x10-23.
Fine-structure constant, α. Straight from wikipedia, though I re-typed the formulae:
Three equivalent definitions of α in terms of other fundamental physical constants are:
where:
- 1.•e is the elementary charge;
- 2.•ħ = h/2π is the reduced Planck constant;
- 3.•c is the speed of light in vacuum;
- 4.•ε0 is the electric constant or permittivity of free space;
- 1.•or µ0 is the magnetic constant or permeability of free space;
- 2.•or ke is the Coulomb constant.
- 1.•or µ0 is the magnetic constant or permeability of free space;
- 5.In electrostatic cgs units, the unit of electric charge, the statcoulomb, is defined so that the Coulomb constant, ke, or the permittivity factor, 4πε0, is 1 and dimensionless. Then the expression of the fine-structure constant becomes the abbreviated α = e2/ħc which is an expression commonly appearing in physics literature.
In natural units, commonly used in high energy physics, where ε0 = c = ħ = 1, the value of the fine-structure constant is α=e2/4π. As such, the fine-structure constant is just an alternative expression of the elementary charge; e=√4πα ≈0.30282212 in terms of the natural unit of charge.
α = 7.2973525698x10-3. Often used instead is α-1 = 137.035999074.
Exercise:
- 1.Light speed is 3x1010m/s. Walking at 5kph is what in m/s? And as a decimal part of c? Now check the speed of sound 330m/s in terms of c.
- 2.Give your own definitions for Boltzmann and light. And then look them up properly.
- 3.Using Planck units, convert (i) g=9.807 m/s/s (ii) the speed of sound at 330m/s
(iii) atmospheric pressure of 1000 millibars (that’s one bar, surely?) (iv) you pick something different from these three.