Learning Scientific Programming with Python (2nd edition)
P6.5.1: Planck units of measurement
Question P6.5.1
In physics, the Planck units of measurement are those defined such that the five universal physical constants $c$ (the speed of light), $G$ (the gravitational constant), $\hbar$ (the reduced Planck constant), $(4\pi\epsilon_0)^{-1}$ (the Coulomb constant) and $k_\mathrm{B}$ (the Boltzmann constant) are set to unity. The dimensions of these quantities in terms of length (L), mass (M), time (T), charge (Q) and thermodynamic temperature ($\mathrm{\Theta}$) are given in the table below, along with their values in SI units.
| Constant | Value | Dimensions | |
|---|---|---|---|
| $c$ | Speed of light | $2.99792458 \times 10^8 \;\mathrm{m\,s^{-1}}$ | $\mathrm{L\,T^{-1}}$ |
| $G$ | Gravitational constant | $6.6743\times 10^{-11} \;\mathrm{m^3\,kg^{-1}\,s^{-2} }$ | $\mathrm{L^3\,M^{-1}\,T^{-2}}$ |
| $\hbar$ | Reduced Planck constant | $1.0545718176461565\times 10^{-34} \;\mathrm{J\,s}$ | $\mathrm{L^2\,M\,T^{-1}}$ |
| $(4\pi\epsilon_0)^{-1}$ | Coulomb constant | $8.987551786170798 \times 10^9 \;\mathrm{N\,m^2\,C^{-2}}$ | $\mathrm{L^3\,M\,T^{-2}\,Q^{-2}}$ |
| $k_\mathrm{B}$ | Boltzmann constant | $1.380649 \times 10^{-23} \;\mathrm{J\,K^{-1}}$ | $\mathrm{L^2\,M\,T^{-2}\,\Theta^{-1}}$ |
This suggests the following matrix relationship between the constants and their dimensions:
$$ \begin{array}{crrrrr} {}&\mathrm{L} & \mathrm{M} & \mathrm{T} & \mathrm{Q} & \mathrm{\Theta}\\ c & 1 & 0 & -1 & 0 & 0\\ G & 3 & -1 & -2 & 0 & 0\\ \hbar & 2 & 1 & -1 & 0 & 0\\ (4\pi\epsilon_0)^{-1} & 3 & 1 & -2 & -2 & 0\\ k_\mathrm{B} & 2 & 1 & -2 & 0 & -1 \end{array} $$
Using the inverse of this matrix, determine the SI values of length, mass, time, charge and temperature in the base Planck units; that is, the combination of these physical constants yielding the dimensions L, M, T, Q and $\mathrm{\Theta}$. For example, the Planck length is found to be $l_\mathrm{P} = \sqrt{\hbar G / c^3} = 1.616199 \times 10^{-35}\;\mathrm{m}$.