The Quantum Classical Clock
This clock is a simulation of a quantum classical device. We usually think of classical mechanics as the limit of quantum mechanics in the large, when the particles get large or the number of particles get large. But there are also quantum systems which behave classicly at integer times on integer locations and while remaining quantum in behavior at all other locations and times.
The sinc function, sinc(x) = sin(x)/x or 1 when x is 0.
The sinc function squared
The positional basis function, e ^ (i pi x) * sinc(pi x), for large N.
The positional basis function for large N squared.
The positional basis function, (sin pi (u-m) / (N sin pi ((u-m)/N))) * e ^ ((i pi(u-m)) (1-1/N)), for finite N.
The positional basis function for finite N squared.
The positional basis function for finite N (reds) and for large N (blues) compared.
The wave function for a bit located at non-integral locations can be
expressed as a linear combination of the positional basis functions
for integral locations.
The coefficient for the positional basis centered at m in the wave
function centered at t is given by the value of the wave function
centered at t at m. This is true for wave functions centered at
integers, since only one basis function has a non-zero value at the
integer center, and that value is 1.
The overlaps between the 60 positional basis functions for N=60 and fractionally shifted copies between 0 and 1. (This takes a while to compute and delays the loading of this page if inlined, so it's linked to as a standalone svg source.) The overlaps between integral positional basis functions and fractional positional functions between 0 and 1.
Last modified: Mon Aug 25 20:17:00 MDT 2003 elf.org