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Atomic units

In atomic and molecular physics, it is convenient to use the elementary charge, e, as the unit of charge, and the electron mass, tex2html_wrap_inline120 , 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 tex2html_wrap_inline126 , which has dimensions tex2html_wrap_inline128 , and another quantity that appears all over in quantum physics is tex2html_wrap_inline130 Planck's constant divided by tex2html_wrap_inline132 , which has dimensions tex2html_wrap_inline134 ; so it is convenient to choose units of length and time such that tex2html_wrap_inline136 and tex2html_wrap_inline138 .

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 tex2html_wrap_inline144 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 tex2html_wrap_inline146 ). The unit of time is tex2html_wrap_inline148 .

A fuller list of atomic units and their precise values is PQRG, page 51. It is worth noting that the unit of velocity is tex2html_wrap_inline150 tex2html_wrap_inline152 , which can be written as tex2html_wrap_inline154 , 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 tex2html_wrap_inline160

Leaving out the correction factor tex2html_wrap_inline162 , the radius of the H atom is 1 bohr = 0.0529 nm, the speed of the orbiting electron is tex2html_wrap_inline166 and the ground state energy is tex2html_wrap_inline168 (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 tex2html_wrap_inline170 and tex2html_wrap_inline172 .


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 tex2html_wrap_inline140. 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 tex2html_wrap_inline140 .

next up previous
Next: Astronomical units Up: Choice of units Previous: Choice of units

Vittorio Celli
Thu Sep 4 00:46:12 EDT 1997
======== DISCARDED proton mass, tex2html_wrap_inline122 , or the