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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.

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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.

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