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Does the following pair of probability distributions look familiar to anybody?

They are arrived at via simulating the following toy quantum model. There is a single particle, and it can be anywhere in a 1-dimensional world. It can furthermore be in one of two states, call them L and R. Each timestep, execute the following transition function: an L-particle moves to the left one discrete hop, say of distance epsilon, and becomes the superposition (|L>+|R>)/√2. An R-particle moves to the right one epsilon-hop, and becomes the superposition (|L>-|R>)/√2. Notice the minus sign! Exercise: convince yourself this evolution is unitary.

The graph above is the probability of finding an L-particle (red line) or an R-particle if we do this process 250 times starting with an L-particle at 0.5, with epsilon chosen so that 250 hops to the left is 0, and 250 hops to the right is 1. Notice how the green distribution seems to be symmetric. I don't know why that is, or how to prove that it is.

Notice also that the most likely place to find the particle isn't near where it started, and it also isn't at 0 or 1 --- instead, it seems to want to travel at 1/√2 of the "speed of light" --- probably to the left and stay an L-particle, but maybe to the right and become an R particle. If we had started with an R particle, the graph would come out the same except with red exchanged for green, and the whole thing mirror-reversed horizontally. Zooming in at around (1/√2 + 1)/2 ~= 0.8535... and cranking up the number of iterations to 2500, we see a nice wibbly peak:

Anyway, this distribution is mysterious to me and I don't know of any closed form for it. I wonder if it's a well-known toy model among physicists?

They are arrived at via simulating the following toy quantum model. There is a single particle, and it can be anywhere in a 1-dimensional world. It can furthermore be in one of two states, call them L and R. Each timestep, execute the following transition function: an L-particle moves to the left one discrete hop, say of distance epsilon, and becomes the superposition (|L>+|R>)/√2. An R-particle moves to the right one epsilon-hop, and becomes the superposition (|L>-|R>)/√2. Notice the minus sign! Exercise: convince yourself this evolution is unitary.

The graph above is the probability of finding an L-particle (red line) or an R-particle if we do this process 250 times starting with an L-particle at 0.5, with epsilon chosen so that 250 hops to the left is 0, and 250 hops to the right is 1. Notice how the green distribution seems to be symmetric. I don't know why that is, or how to prove that it is.

Notice also that the most likely place to find the particle isn't near where it started, and it also isn't at 0 or 1 --- instead, it seems to want to travel at 1/√2 of the "speed of light" --- probably to the left and stay an L-particle, but maybe to the right and become an R particle. If we had started with an R particle, the graph would come out the same except with red exchanged for green, and the whole thing mirror-reversed horizontally. Zooming in at around (1/√2 + 1)/2 ~= 0.8535... and cranking up the number of iterations to 2500, we see a nice wibbly peak:

Anyway, this distribution is mysterious to me and I don't know of any closed form for it. I wonder if it's a well-known toy model among physicists?