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 .
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 .