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).

  • $\begingroup$ The initial positions of what at the beginning of the universe ? You might also consider this in relation to the answer to Did the Big Bang happen at a point ? $\endgroup$
    – StephenG
    Oct 6 '18 at 6:05
  • $\begingroup$ I mean the initial positions of "particles" trajectories of which is concerned in Bohmian mechanics. I think they don't talk about big bang. In Bohmian mechanics, they say that the appearance of randomness in the outcome of experiments is due to randomness in this initial positions. Their uncertainty (randomness) is proportional to psi^2 (born's rule). Now I want to know how psi^2 evolves with time. Is a Gaussian diatribution, for example, broaden as time passes? $\endgroup$ Oct 7 '18 at 7:16

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

  • $\begingroup$ Thanks. The reason I asked about the amount of the uncertainty was a problem I have with quantized spacetime. Suppose that you have a gaussian wavefunction for a particle. As time goes forward, the wavefunction will spread. So as we go backwards in time, it will get narrower and more localized. how much can we make the gaussian distribution narrower? In the case that the space is quantized like a grid in the plank scale, I think it is meaningless to make the gaussian distribution narrower than a quanta of space, because in that scale just certain positions are possible. $\endgroup$ Oct 15 '18 at 22:48

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