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 clock rendered in SVG using a standards compliant browser. The sinc function, sinc(x) = sin(x)/x or 1 when x is 0. The sinc function rendered from -3 to +3. The sinc function squared The sinc function squared rendered from -3 to +3. The positional basis function, e ^ (i pi x) * sinc(pi x), for large N. The positional basis function for large N rendered from -3 to +3. The positional basis function for large N squared. The positional basis function for large N squared rendered from -3 to +3. 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 rendered from -3 to +3. The positional basis function for finite N squared. The positional basis function for finite N squared rendered from -3 to +3. The positional basis function for finite N (reds) and for large N (blues) compared. 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. Table of Coefficients. 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.
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