Heisenberg uncertainty principle for macroscopic objects

[Ben Crowell <bcrowell@SPRINTMAIL.COM>]

I wrote:
> Well, without worrying about any of the niceties, the de Broglie
> wavelength of the elephant is surely extremely short

Leigh replied:
> ...and meaningless. The niceties are *very* important. Just because
> you can plug the mass and velocity of an elephant into a formula
> you should not conclude that you have inferred something physically
> relevant by doing so.

Leigh, you are of course correct that I was making a questionable
leap of intuition. On the other hand, I would be interested in
understanding why coherence is so relevant. Here is a "straw man"
calculation that may be right or may be wrong, but that I hope
gets deeper into the issue. Let's compare two different ways of
calculating the tunneling probability:

(1) If the elephant behaved just like a lead nucleus or a buckyball,
then the tunneling probability would be
 P = exp[ -x sqrt(2 mtotal Vtotal) / hbar]
where
 x = thickness of wall
 mtotal = total mass of our particle-like elephant
 Vtotal = total amount by which the elephant's energy
  falls below the classical minimum for getting through
  the wall
Letting n be the total number of atoms in the elephant, we have
 mtotal = n matom
 Vtotal = n Vatom
and the probability becomes
 P = K^n
where
 K = exp [-x sqrt(2 matom Vatom)/hbar)
is the probability for a single atom to tunnel through the wall.

(2) Now suppose the elephant is completely incoherent. We treat
the tunneling of each atom as if it was a completely independent
process. We have n events, each with probability K, so the probability
that all the events will occur independently is
 P = K^n

Now the fact that the two calculations come out the same would seem
to argue against any important dependence on coherence or incoherence.
However, I'd guess that my calculation is wrong because it implicitly
assumes that all the atoms make their assault on the barrier
simultaneously
and that if they do all get to the other side, they nicely reassemble
into an elephant. In nuclear physics, for instance, one typically
discusses alpha decay in terms of a tunneling probability multiplied
by a probability of preformation of an alpha-particle cluster within
the parent nucleus. (This is not an entirely meaningful concept, since
the existence of true alpha particles inside a nucleus generally
violates the Pauli exclusion principle, but anyway it's the way people
think about it.)

I would guess that the expectation value of the amount of elephant-
mass on the far side of the wall is the same regardless of whether
it's a coherent elephant (which clearly doesn't exist!) or an
incoherent elephant. But the probability that that mass assembles
itself into an elephant is probably zillions of orders of magnitude
less for the incoherent elephant.

We should also probably refine our terms and instead of talking
about tunneling probabilities for microscopic objects versus
macroscopic objects, we should talk about tunneling probabilities
for coherent macroscopic objects versus incoherent macroscopic
objects. Coherent macroscopic objects do exist, e.g. the Bose-
Einstein condensate.

Anyway, if I'm a mindless plug-and-chugger, I'm glad that I did the
same mindless plug-and-chug as Gamow!

 Ben Crowell
 Fullerton College