In Bohmian mechanics, the initial configurations and the pilot wave determine the future of the system. Given a set of initial positions for particles and a set of arbitrary trajectories, can we define a pilot wave that guides the particles on that trajectories?
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2 Answers
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A funny question which has not really much meaning for physics, but serves more as a mathematical puzzle, I guess.
Let's look at the guiding equation for the configuration $Q \in \mathbb{R}^{3N}$ (without spin for simplicity, $\hbar=2m=1$):
$$ \frac{dQ(t)}{dt} = \Im \frac{\psi^* \nabla \psi}{|\psi|^2}(Q(t)) $$
Now, given an arbitrary differentiable (this at least should hold!) trajectory $Q(t)$, the mathematical question you ask is if there exists a function $\psi: \mathbb{R}^{3N} \to \mathbb{C}$ such that this equation is satisfied. I can't provide a proof here, but you can see that the left hand side is given and the right hand side is thus determined only at the trajectories, so there is a large freedom and there should be a large number of possible choices for $\psi$ that work.
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$\begingroup$ I was thinking that if we can produce all possible trajectories by modifying the guiding wave, this will be another way to produce the quantum randomness. I mean, in Bohmian mechanics, we assume both the guiding wave and initial positions as something random and then the system evolves according to them deterministically. We usually assume that the guiding wave is fixed and different initial positions lead to different outcomes in quantum measurements, but can we assume the initial positions to be fixed and then define different futures for the system with different guiding waves? $\endgroup$– Alex LCommented Nov 23, 2018 at 21:13
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$\begingroup$ In an experiment, you can control the initial wave function, but not the initial positions, that's the whole point. You can make many runs of the experiment with always (more or less) the same wavefunction, but not with the same initial positions. Thus, the statistical behaviour of measurement results must be explained by different initial positions. $\endgroup$– LukeCommented Nov 26, 2018 at 9:50
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$\begingroup$ Thanks. I have another question: In the real world, how are the trajectories we see? Are they always ordered with definite shapes, or they are kind of disordered and random-shaped (or both)? Is their shape merely defined by the guiding wave? $\endgroup$– Alex LCommented Nov 29, 2018 at 20:40
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It may be that there are possible integral curves obtainable from the guiding wave equation showing arbitrary trajectories to allow for quantum randomness, but this would be purely mathematical. Even in standard quantum theory, we can obtain solutions to the Schrödinger equation, would be wave functions, that are physically impossible and are discarded on that basis.
This is more pronounced in Bohmian Mechanics which is a realistic theory where the mathematics describes physical processes; including that the wave function describes the pilot wave. For a system the wave function can be known precisely and does not change state, (unless a measurement is made). This must apply to the pilot wave also. Each particle in a system is represented by the same wave function and thus guided by an identical pilot wave.
With the wave function known precisely the possible particle positions will show a probabilistic distribution according to the wave function squared. But we have no way of knowing a particle’s position within that distribution. If we take a measurement to find out, the wave function collapses to a different state and so we would now have a different pilot wave but for a different system, not the one under consideration. So while the initial particle positions are random the pilot wave is not.
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$\begingroup$ First, I think the initial pilot wave is also considered to be random, if not, what determines it? As you pointed out, in Bohmain mechanics we produce the quantum randomness using a fixed guiding wave function and random initial positions. Yes, this is what they do in the conventional approach. I aimed to figure out if another option is possible, that we fix the initial positions and try to produce different outcomes (quantum randomness) by tuning the guiding wave. $\endgroup$– Alex LCommented Nov 25, 2018 at 20:46
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$\begingroup$ You told this is mathematically possible, but why shouldn't it be physically possible? We are only trying to produce different outcomes for experiments by changing the wavefunction. by producing "arbitrary trajectories" I didn't mean to produce strange and physically impossible future states for the system. So I think producing "Qunatum randomness" (which deals with randomness in the micro scale) should be physically possible. $\endgroup$– Alex LCommented Nov 25, 2018 at 20:51
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$\begingroup$ Luke's comment about the experimenter being able to control the wave function is very good. My additional point is that any change to that wave function, come pilot wave, will result in a new set of random trajectories, which cannot provide any explanation for the set of trajectories we were considering. For example, in the two-slit experiment the wave form is simply passing through both slits & interfering. If we measure a particle to find which slit it trajected, the waveform does change but contains a different set of random trajectories, corresponding to the wave passing through one slit. $\endgroup$ Commented Nov 26, 2018 at 15:23