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6.055J / 2.038J The Art of Approximation in Science and Engineering
Spring 2008
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6.055 / Art of approximation
54
Whereas using dimensions reveals that the problem is simpler: to find the function
h
such
that
kT
2
=
h
()
.
m
Here
h
()
means a function of no variables. Why no variables? Because the right side con
tains all the other quantities on which
kT
2
/
m
could depend. However, dimensional analysis
says that the variables appear only through the combination
kT
2
/
m
, which is already on the
left side. So no variables remain to be put on the right side; hence
h
is a function of zero
variables. The only function of zero variables is a constant, so
kT
2
/
m
=
C
.
This pattern illustrates a famous quote from the statistician and physicist Harold Jeffreys
[
6
, p. 82]:
A good table of functions of one variable may require a page; that of a function of
two variables a volume; that of a function of three variables a bookcase; and that of
a function of four variables a library.
Use dimensions; avoid tables as big as a library!
7.4 Hydrogen atom
Hydrogen is the simplest atom, and studying hydrogen is the simplest way to understand
the
atomic theory
. Feynman has explained the importance of the atomic theory in his
famous lectures on physics [
7
, p. 12]:
If, in some cataclysm, all of scientific knowledge were to be destroyed, and only
one sentence passed on to the next generations of creatures, what statement would
contain the most information in the fewest words? I believe it is the
atomic hypothesis
(or the atomic
fact
, or whatever you wish to call it) that
all things are made of atoms
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 Spring '08
 SanjoyMahajan
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