Posted by John Stevenson on 06/10/2016 22:10:39:
Or hpw many angels can dance on a photo copied page ?
Perhaps that should be a mantra for this forum ?
Well it's easy enough to calculate, all you need to do is this, and multiply the result by the number of pinheads you can fit on a sheet of copy paper:
Assuming that each angel contains at least one bit of information (fallen / not fallen), and that the point of the pin is a sphere of diameter of an Ångström (R=10exp-10 m) and has a total mass of M=9.5*10exp-29 kilograms (equivalent to that of one iron atom), we can use the Bekenstein bound[3] on information to calculate an upper bound on the angel density. In a system of diameter D and mass M, less than kDM distinguishable bits can exist, where k=2.57686*10exp43 bits/meter kg.[7] This gives us a bound of just 2.448*10exp5 angels, far below the Schewe bound.
Note that this does not take the mass of angels into account. A finite angel mass-energy would increase the possible information density significantly. If each angel has a mass m, then the Bekenstein bound gives us N<kD(M+Nm). Beyond mcrit>1/kD �3.8807*10exp-34 kg this produces an unbounded maximal angel density as each angel contributes enough mass-energy to allow the information of an extra angel to move in, and so on.
However there is uncertainty about the extra space needed to allow for certain dance steps, and uncertainty about the effects that quantum -speed dancing may have on the base material, more so for paper than for pins in this case. So:
The uncertainty relation also imposes a limitation on the dance. Since the uncertainty in position of the angels by assumption is less than the size of the point �x�R we find that the uncertainty in momentum must be �p�hbar/R, and this leads to a velocity uncertainty �v>hbar/Rm. If m= mcrit we get �v>> 8.6766*10exp59 m/s (>> c), which shows that:
(1) the angels must dance with speeds near the velocity of light in order to obey quantum mechanics;
(2) a full relativistic treatment is necessary; and
(3) that the precision of the dance must break down due to quantum effects.
This can be used to rule out certain types of dance due to their high precision requirements.