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Some Useful Number and Unit Combinations
As we begin doing quantitative calculations in atomic physics, we quickly find that the “normal” SI quantities for
things like length, energy, etc., cause us constantly to be dealing with large negative numbers as powers of 10, which can lead to
incorrect results on the calculator. It is better if we can work in so-called “natural units”, where length is expressed in atomic
dimensions like nm instead of meters and energy is in eV instead of Joules. In this exercise, you’ll convert a couple of very
important constants and combinations of constants to “natural” units. Here are some important numbers you’ll need.
Planck’s constant
speed of light
electronic charge
mass of electron
electric force constant
length conversion
energy conversion
h = 6.626 x 10-34 J-s
h = h/2
c = 2.998 x 108 m/s
e = 1.602 x 10-19 C
me = 9.109 x 10-31 kg
kE = 8.988 x 109 N-m2/C2
1 nm = 10-9 m
1 eV = 1.602 x 10-19 J
Example:
Express Planck's constant in eV-s
(6.626 x 10-34 J-s) (1 eV/1.602 x 10-19 J) = 4.136 x 10-15 eV-s
(This still has an uncomfortable exponent. The other examples you'll work out will give much nicer results. This calculation is
just shown as an example.)
1.
The combinations hc and hc occurs quite often in atomic physics. Show that these have units of energy x length, and
express them in units of eV-nm.
2.
Another energy x length combination of fundamental quantities is kEe2. Please express this in eV-nm.
Atomic1 - 1
 Stephen Luzader
3.
It is often useful to remember the c, the speed of light in empty space, can be used as a unit. It can be combined with
the mass of an object to express that mass as an energy. For example, a combination that appears often in atomic physics is
h2/me , where me is the mass of the electron. In SI units, this is 9.1 x 10 -31 kg, which is an ugly number to punch into a
caclulator. However, the energy equivalent is easier to work with. In eV, mec2 = 511 x 103 eV (511 keV). Watch how we can
use this together with the results of exercise 1 to turn this into something relatively easy to work with:
h
2
me
2 2

h c
me c
2

(1240 eV - nm)
2
 3.01 eV - nm
2
511000 eV
We multiplied the numerator and denominator by c2 to create combinations of symbols that can be interpreted easily in "natural
units". For practice, calculate the energy of the first Bohr orbit entirely in natural units, following this example.
Atomic1 - 2
 Stephen Luzader