Planck Units - Planck Units Simplify Key Equations

Planck Units Simplify Key Equations

Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (1 second is not the same as 1 metre). In theoretical physics, however, this scruple can be set aside, by a process called nondimensionalization. Table 4 shows how Planck units, by setting the numerical values of five fundamental constants to the number one, nondimensionalizes and simplifies many fundamental equations of physics.

Table 4: How Planck units simplify the key equations of physics
Usual form Nondimensionalized form
Newton's law of universal gravitation
Einstein field equations in general relativity
Mass–energy equivalence in special relativity
Energy–momentum relation
Thermal energy per particle per degree of freedom
Boltzmann's entropy formula
Planck's relation for energy and angular frequency
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T.
Stefan–Boltzmann constant σ defined
Bekenstein–Hawking black hole entropy
Schrödinger's equation - \frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \hbar \dot{\psi}(\mathbf{r}, t) - \frac{1}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \dot{\psi}(\mathbf{r}, t)
Hamiltonian form of Schrödinger's equation
Covariant form of the Dirac equation
Coulomb's law
Maxwell's equations





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