From: jbaez@euclid.mit.edu (John C. Baez)
Subject: Length scales in physics 1 - the Bohr radius
Organization: MIT Department of Mathematics, Cambridge, MA
Distribution: na
Date: Mon, 17 Aug 92 20:11:59 GMT

Dilation invariance is also called scale-invariance, and it's implied by
another popular property known as conformal invariance.  Jargon aside,
it's the property of laws of physics that look the same no matter
what length scale you examine.  Most of the world around us
follows laws that are NOT dilation-invariant.  Instead, the laws of
physics contain various built-in units of length.  That's why there are no
ants 60 feet tall, or atom-sized elephants.   As I mentioned in my post
on symmetry, LIGHT is one of the few things around that manifests
dilation invariance.

I was going to post on dilation invariance but I got distracted by
trying to figure out some physical length scales from scratch.  So I
am going to post 4 separate articles, one on each of 4 length scales
that are important in 4 different domains of physics.

1.   The Bohr radius

Physics is full of characteristic length scales: the Bohr radius of
the hydrogen atom, the Compton wavelength of the electron and the classical
electron radius being three.  I want to start by showing how the Bohr radius,
which is the length scale of atomic physics, depends on the electron
mass, the electron charge, and Planck's constant.  In particular I
want the reader to see an example in which a mass (the electron mass)
is proportional to the *inverse* of a length (the Bohr radius).  It is
an example of a general phenomenon: a mass scale sets an inverse
length scale.  

The Bohr radius r is the approximate size of a hydrogen atom in its
ground state  - approximate in that the electron in a hydrogen atom is a
fuzzy probability cloud, not a pellet whizzing around in a circular
orbit!  I forget what the Bohr radius equals, so I'll have to work it out.  

First I will cheat and figure it out using dimensional analysis.  (I
want to prove to the real physicists there that I can do this stuff
that mathematicians are usually scared to do.)  We'll just get an
order-of-magnitude estimate of the Bohr radius, modulo factors of 4pi^2
and such.  So what could the hydrogen atom's radius depend on?  It's
two particles of equal charge attracted by the electric field, and it
is kept from collapsing by quantum effects, namely the uncertainty principle.
Since it involves a charged electron whizzing around, it'll depend on the 
electron charge e and the electron mass m.   Since the electron is in a
fuzzed-out orbit because of quantum effects, it'll involve Planck's
constant hbar. Since the proton is 1836 times as heavy as the electron,
one can be approximate it quite well by being fixed, so the Bohr radius
WON'T depend much on the proton mass.  (We are making an approximation
here; a better approach uses the center of mass frame and our approach
will be off by terms proportional to the mass of the electron over the
mass of the proton.)  This is not a relativistic calculation so we won't
use c.  

In short, the answer will probably involve:

e - the electron charge
m - the electron mass
hbar - Planck's constant

and nothing else.  Okay.  So using dimensional analysis, what can do
with these to get a quantity with dimensions of length?  e has dimensions 
of charge.  If we absorb Coulomb's constant in the definition of charge,
charge has dimensions of length times force^{1/2}, since

force = charge^2/distance^2,  

by Coulomb's law.

Thus e has dimensions L(ML/T^2)^{1/2} = L^{3/2}M^{1/2}/T,
     m has dimensions M,
  hbar has dimensions momentum times distance = ML^2/T.

To combine these to get a length, we fiddle around.  Note that hbar^2/e^2
has dimensions ML, so hbar^2/me^2 has dimensions L.

Therefore the Bohr radius is 

r = hbar^2/me^2 

times some dimensionless constant.   Cheating, I peek in my uncle's
old physics text, The New College Physics - A Spiral Approach - since
I've packed up all my other books.  The dimensionless constant turns out to
be 1/4pi^2, and the Bohr radius works out to be 5 times 10^{-11} meters.
Half an angstrom, in other words.  Angstroms are the length scale of
atomic physics.

Note that 

r = hbar^2/me^2 

makes some sense in that it gets bigger when hbar gets bigger (more
"quantum fuzziness" makes a bigger atom) and gets smaller when the electron's
charge is bigger (more attractive pull).  It also gets smaller when the mass
of the electron gets bigger - here we see that bigger mass scales go
with smaller length scales!  But we want some understanding at a gut level 
why making the electron *heavier* would make the hydrogen atom *smaller*!  
It's known that this is true, by the way, because one can take a muon,
which is just like an electron but 206.77 times heavier (and decays
rapidly), and make muonic hydrogen which is about 206.77 times smaller.
But WHY? 

Here's a very shoddy "derivation" of the Bohr radius that
makes it clear why big electron mass would give small atoms!  The
uncertainty principle says that the product of the uncertainty of
momentum and the uncertainty of position can't be too small:

Delta p Delta x >= hbar

(leaving out factors of 2 etc.).  Now in the hydrogen atom Delta x is
just the Bohr radius and we can write Delta p, the uncertainty in
momentum, as  m Delta v, where v is velocity.  So we get

r = hbar/m Delta v.						1)

Note that I have replaced an inequality by an equality!  That is
because this whole argument is so bogus that it doesn't matter - all
the = signs below should be read as "is approximately equal to
[hopefully]".   Mathematicians should learn to respect this sort of
vague argument that physicists are so good at - one can often squeeze
the right answer out of very sketchy reasoning!

We see here why bigger m give smaller Bohr radius r: the uncertainty
principle allows us to "squish up" the wavefunction of a more massive
particle in a smaller region of space with the same Delta v.  But how
big is Delta v?  To complete this shoddy "derivation" of Bohr's
radius, note that Delta v is probably about equal to the velocity of a
classical electron going around a proton in an orbit the size of the 
Bohr radius.  How do we relate the velocity of an orbiting electron
and the radius of its orbit?   We can get an approximate answer by
simply equating the kinetic and potential energy

m(Delta v)^2/2 = e^2/r						2)

since the virial theorem says kinetic and potential energies are about
the same magnitude.  Now squaring 1) we get

(Delta v)^2 = hbar^2/m^2 r^2					3)

and dividing 2) by 3) we get

m = e^2 m^2 r/hbar^2						4)

or just

r = hbar^2/me^2!

I have emphasized by the use of words such as "shoddy" that this is a
rough heuristic argument and not at all a mathematical derivation of
the Bohr radius.  Indeed, no matter how sloppy our argument was it was
bound to give the same answer as long as it was DIMENSIONALLY correct.
This should make one distrust the argument!   But one can make it utterly
precise, as Schrodinger did.  We see at any rate that the Bohr radius 
can be guessed by essentially classical reasoning together with the
uncertainty principle, and this length scale is naturally proportional
to an inverse mass scale - the inverse of the electron mass!