When I awoke from troubled dreams this morning I found myself thinking about the discrete fourier transform. If you take a signal, and phase-shift each frequency component by some phase proportional to the square of its frequency, shouldn't that give you a kind of Laplace operator? The reasoning being that this is like saying that energy is proportional to the square of momentum, so the the time derivative of the result should look like the square of the momentum operator, hence a second spatial derivative. But if I do this with discrete time and space, it's not really a derivative, just some funny analogue of it.
In mathematica I can do
Delta[x_, y_] = If[x == y, 1, 0]; M = 100; K = 100;
H[s_] = s^2/(2 M K);
f[x_] = (1/(2 M)) Sum[
Exp[2 I Pi (x s - H[s])/(2 M)], {s, -M, M - 1}];
rt = Table[{x, K*K*(Re[f[x]] - Delta[x, 0])}, {x, -20, 20}];
st = Table[{x, Re[f[x]] }, {x, -20, 20}];
it = Table[{x, K *Im[f[x]]}, {x, -20, 20}]; ListPlot[{st , it, rt },
Joined -> True, Filling -> Axis, PlotRange -> (1 {-0.6, 1})]
and it yields this graph
which seems to be convergent as I increase M, the number of points in the world, and K, a scaling factor that when bigger makes every momentum count as less energetic. Maybe it is effectively acting like hbar (or its reciprocal) or something? I dunno.
Doing some numerical fiddling, it seems the spikes in the imaginary part (the red graph) just one step away from the origin on either side have the value of exactly 1/π --- not sure what causes that. |
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