Murray Gell-Mann on Entanglement

[MrRobotoToo]

There is another big difference with diffusion, and that is that diffusion is a matter of some substance spreading out in physical space, while a wave function propagates in configuration space. The difference isn't apparent when you're talking about a single particle, but becomes important when you are talking about multiple particles. For two particles, the wave function is a function of 6 variables: $\psi(x_1, y_1, z_1, x_2, y_2, z_2)$ where $(x_1, y_1, z_1)$ refers to the location of the first particle, and $(x_2, y_2, z_2)$ refers to the location of the second particle. Because it's a function of configuration space, there is no meaning to "the value of the wave function here". So, in spite of the similarity of form, the Schrödinger equation is nothing like a diffusion equation (at least not diffusion through ordinary 3-space).

Good points. I would just point out that even in the multiparticle case it would still make sense to draw an analogy with a random walk. In the latter case we would want to calculate $P\left(x_{1},y_{1},z_{1},x_{2},y_{2},z_{2}\right )$, i.e. the probability of finding the particles at $\vec{x}_{1}$ and $\vec{x}_{2}$ after the initial system preparation. The Feynman diagrams for two particles would have a natural translation into a random walk analysis for two classical particles.

 

[stevendaryl]

Good points. I would just point out that even in the multiparticle case it would still make sense to draw an analogy with a random walk. In the latter case we would want to calculate $P\left(x_{1},y_{1},z_{1},x_{2},y_{2},z_{2}\right )$, i.e. the probability of finding the particles at $\vec{x}_{1}$ and $\vec{x}_{2}$ after the initial system preparation. The Feynman diagrams for two particles would have a natural translation into a random walk analysis for two classical particles.

Right, but in classical probability theory (with no nonlocal interactions), the probabilities for random walks factor for particles that are too far apart to interact. That is, for two particles that are far apart,

$P(x_1, y_1, z_1, x_2, y_2, z_2) \approx P(x_1, y_1, z_1) P(x_2, y_2, z_2)$

The random walk taken by this particle is independent of the random walk taken by this other particle. If that fails to be the case, then one suspects that there is some unaccounted-for long range interaction or shared state.