The intuition:
For a high dimensional ball, most of the volume is near the surface, and most of the surface is near the equator (for any given choice of equator). The extremity of “most” and “near” increases with number of dimensions. The intersection of two equal-size balls is a ball minus a slice through the equator, and thus missing most of its volume even if it’s a pretty thin slice.
The calculation:
Let
%20=%202\int%20_{y=0}%5E{r}%20v(n-1,\sqrt{r%5E2-y%5E2})%20dy%20=%20(2%20\pi%5E{n/2}%20r%5En)%20/%20(n%20\Gamma(n/2))) which is the volume of a n-dimensional ball of radius r. Then the fraction of overlap between two balls displaced by x is
%20dy}{v(n,r)}) (The integrand is a cross-section of the intersection (which is a lower-dimensional ball), and y proceeds along the axis of displacement.) Numeric result.
The position of the apparatus has to be uncertain enough for you to be able to measure momentum (i.e. acceleration) precisely enough. It works out just fine to patterns being smeared, an interesting exercise to do mathematically though.
edit: didn’t see context, thought you were speaking of the regular double slit experiment. It still applies though.
With regards to the M1 I don’t quite understand the question as the spin is not an arrow that snaps from arbitrary orientation to parallel or anti-parallel. When it interacts with field, after the speed of light lag, there’s recoil.
The intuition: For a high dimensional ball, most of the volume is near the surface, and most of the surface is near the equator (for any given choice of equator). The extremity of “most” and “near” increases with number of dimensions. The intersection of two equal-size balls is a ball minus a slice through the equator, and thus missing most of its volume even if it’s a pretty thin slice.
The calculation: Let
Then the fraction of overlap between two balls displaced by x is
Thanks for both the math and the intuitive explanation. Now I’m really curious what the right answer is to the physics question...
The position of the apparatus has to be uncertain enough for you to be able to measure momentum (i.e. acceleration) precisely enough. It works out just fine to patterns being smeared, an interesting exercise to do mathematically though.
edit: didn’t see context, thought you were speaking of the regular double slit experiment. It still applies though.
With regards to the M1 I don’t quite understand the question as the spin is not an arrow that snaps from arbitrary orientation to parallel or anti-parallel. When it interacts with field, after the speed of light lag, there’s recoil.