In atomic and molecular physics, it is convenient to use the elementary
charge, e, as the unit of charge, and the electron mass, 
, as the
unit of mass (although for some purposes the
unified mass unit, u, is more convenient). Electrostatic forces and
energies in atoms are always proportional to 
,
which has dimensions 
, and another quantity that appears all
over in quantum physics is 
Planck's constant divided by 
,
which has dimensions 
; so it is convenient to choose units of
length and time such that 
and 
.
The atomic unit of length is then (by dimensional analysis alone)
This is called the Bohr radius, or simply the bohr, because in the Bohr model it is the radius of the smallest orbit for an electron circling a fixed proton (see Note).
One can similarly find the unit of time by dimensional analysis. Rather than do this directly, we find first the atomic unit of energy, which is obviously
This is denoted by 
and is called the hartree; it is twice the ground
state energy for an electron circling a fixed proton (the ground state
energy of the H atom is 
). The unit
of time is 
.
A fuller list of atomic units and their precise values is PQRG, page 51. It
is worth noting that the unit of velocity is 
, which can be written as 
,
where c is the speed of light and
is a dimensionless quantity known as the fine structure constant. The value
of c in atomic units is then 
Leaving out the correction factor 
, the radius of the H atom is 1 bohr
= 0.0529 nm,
the speed of the orbiting electron is 
and the ground state
energy is 
(the use of the
rydberg as an energy unit is officially discouraged, but it persists).
Another energy unit in common use is the electron-
volt, which is the energy acquired by an electron falling through the
potential difference of 1 volt: it has its origin in experimental practice
and can also be viewed as a mixed atomic-SI unit. The
connection between eV and atomic energy units is that 
and 
.
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Note: In reality, the proton has a finite mass mp. Thus, in the Bohr model the electron and proton circle each other at a distance 
. In the full quantum theory the
particles do not follow a single orbit, but the most likely electron-proton
distance in the Hydrogen ground state is exactly .