On Fri, Dec 01, 2017 at 11:49:06AM +1100, Bruce Kellett wrote:
> On 1/12/2017 8:57 am, Bruce Kellett wrote:
> >
> > The coin is classical, consisting of something of the order of 10^22
> > atoms. Indeterminacy in position as given by the Heisenberg Uncertainty
> > Principle, is undetectably small.
>
> I think it is worth while to put some (approximate) numbers around this. The
> reduced Planck constant, h-bar, is approximately 10^{-27} g.cm^2/s. The
> Uncertainty Principle is
>
> delta(x)*delta(p) >= h-bar/2.
>
> For a coin weighing approximately 10 g and moving at 1 cm/s, the momentum is
> mv = 10 g.cm/s. Taking the momentum uncertainty to be of this order, the
> uncertainty in position, delta(x) is of the order of 10^{-28} cm. A typical
> atom has a diameter of about 10^{-8} cm, so the uncertainty in position is
> approximately 20 orders of magnitude less than the atomic diameter. That is
> why quantum uncertainties are irrelevant for macroscopic objects.
> Uncertainties do not add up coherently for macroscopic objects --
> macroscopic objects act as a unit, and the HUP is irrelevant, even for small
> coins.
>
> Bruce
The point being that the uncertainty in the coin's initial position is
itself due to the amplification of quantum uncertainty by classical
chaos.
If the uncertainty in initial conditions is reduced by measurement to
something like exp(-λt)w, where w is the coin's thickness, λ the
system's maximal lyapunov exponent and t the time of flight, then the
coin can be treated deterministically, with the outcome of the toss
known once initial conditions specified to that level of accuracy.
But in the general case, the initial conditions are not so precisely
known. With MWI, an observer is in a superposition of many different
(albeit decohered) quantum universes, and no God can point to one of
them and say that is the real world. So the outcome of the coin toss
can be traced back to the effect of quantum fluctuations during the
setup of the experiment.
In terms of Brent's comment about the magician tossing and catching a
coin in a deterministic (to the magician) way, I believe that involves
a slight of hand. The way I remember doing this as a kid is to flick
the coin enough to make it wobble, and appear spinning to someone
else, but not to flip over completely. Maybe there is a way of doing
it with a flipping coin where the number of flips can be
predetermined. We can say that these are techniques that reduce the
intrinsic Lypanunov exponent. Apparently it is much harder to achieve
this result if the coin lands on a hard surface, rather than the
tosser's hand, which is why that is usually insisted upon.
--
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Dr Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellow [email protected]
Economics, Kingston University http://www.hpcoders.com.au
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