Challenge Level

The energy levels of the electrons in a hydrogen atom according to the Bohr model are given by

$$

E_n = -\frac{m_ee^4}{8\epsilon_0^2h^2n^2}\, n=1, 2, 3, \dots

$$

where $m_e$ and $e$ are the mass and charge of the electron, $h$ is Planck's constant and $\epsilon_0$ the permittivity of free space.

This expression draws together many the fundamental constants involved in quantum mechanics. What are all of the S.I. units of these constants?

Using just these constants is it possible to create a dimensionless quantity?

Can you make a quantity with dimensions $[L]$ (length) or $[T]$ (time) using these quantities?