How much is the amount of uncertainty in initial positions in Bohmian Mechanics?

Question

In Bohmian mechanics, it is believed that the randomness (uncertainty, lack of knowledge) which is seen in the outcome of experiments is due to the uncertainty in the initial particle positions at the beginning of the universe. I want to know "how much" uncertainty we have about the initial positions at the beginning of the universe.

For example, if we have a Gaussian distribution in our experiment, according to Schrodinger's equation, is the distribution for the initial position a narrower Gaussian, meaning that we have less uncertainty for the initial positions? I think the minimum uncertainty has to be larger that the quanta of length (plank length, 10^-33 m).

Answer 1

It is a misunderstanding if you think that a number or amount could be given to answer this question. First, in case someone gets that wrong, one should always stress that the positions are always well-defined at any time, the particles are really at some definite position in Bohmian mechanics.

The explanation for apparent randomness in experiments in Bohmian mechanics is analogous to the explanation of randomness in classical mechanics, e.g. if you roll dice. It comes from lack of knowledge about initial conditions. if you roll a die, you do not know basically anything about initial position and momentum, so it is reasonable to assume that all numbers appear with the same probability. You can not reasonably quantify the amount of uncertainty, I suppose.

It is a theorem in Bohmian mechanics that if you know the initial wave-function, you cannot gain more knowledge on the initial positions of particles than the $|\psi|^2$-distribution. This is the decisive difference to classical mechanics and leads to the term absolute uncertainty used in this important paper: https://arxiv.org/abs/quant-ph/0308039