# How do you properly derive atomic units from SI units?

I'm learning atomic units and I'm having trouble figuring out how to derive the atomic units version of energy, time, electric field, magnetic field, etc. For example the Wikipedia page has a list of derived units: https://en.wikipedia.org/wiki/Atomic_units#Derived_atomic_units But I can't figure out where those expressions came from. I have converted the Schrodinger equation into atomic units using the scaling $x = r/a_0$ and that was fairly straightforward once I realized $e^2$ needed to be expressed in CGS units. However, it's not clear to me how to use this scaling to get other derived units.

Though 4 years old, I will share my answer because I recently went through a similar exercise converting from SI units to atomic units.

Note: this only how I do this process, which seems to produce correct answers. I am not claiming to be an authority on the physics reasoning.

Start with the SI unit you are trying to convert to atomic units (length, energy etc.) and combine $$e$$,$$\hbar$$,$$m_e$$, and $$\frac{1}{4\pi\epsilon_0}$$ to create that same unit.

Using [$$e$$]=C, [$$\hbar$$]=J$$\cdot$$s,[$$m_e$$] = kg, [$$\frac{1}{4\pi\epsilon_0}$$] = $$\frac{m}{F}$$ = $$\frac{m\cdot J}{C^2}$$, one will find that:

$$\frac{m_e\cdot e^4}{\hbar^2}(\frac{1}{4\pi\epsilon_0})^2$$ = J

This is the conversion factor for the atomic unit for energy, which we call a Hartree.

Numerically $$\frac{m_e\cdot e^4}{\hbar^2}(\frac{1}{4\pi\epsilon_0})^2$$ = $$4.35974...\cdot10^{-18}$$ J = 1 Hartree.

The process is the same for other units, just find the combination of those 4 constants that give you the unit of interest.