# What are Bohmian trajectories for a free electron?

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• Ali Lavasani
In summary, the wavefunction in Bohmian mechanics is said to be unknowable, but it can be well-shaped.
Ali Lavasani
A free electron, or any other quantum particle, has an uncertain position/momentum, according to Heisenberg uncertainty principle. The squared amplitude of the wavefunction determines the probability of finding the electron at any point of the space. Accordingly, atomic orbitals are attributed certain shapes (like spherical), within which this probability is higher, but they give no information about the exact position of the electron.

However, in the Bohmian interpretation, there is no such thing as inherent uncertainty, i.e. the uncertainty is just because we can't measure the position of the particle without disturbing its wavefunction. In other words, the particle does have a certain position and trajectory, on which it's moving as determined by the wavefunction (guiding wave).

My question is, if the particle (say a free electron, or any electron in an atom) is moving on a certain trajectory in this interpretation, then what is that trajectory like? Is it supposed to have a certain shape, or it may have any random shape, depending on what the pilot wave is?

P.S. Most of the pictures in papers and scientific websites show the trajectories in a double-slit experiment (see the last picture here: http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/whatisbm_pictures_doubleslit.html). These trajectories look very ordered and well-shaped, but does any particle have such ordered trajectories?

Bohmian trajectories are "ordered and well-shaped" as in the pictures that you have seen, as long as the wave functions that guide them are "ordered and well-shaped" too. For intuition, you can think of particle trajectories guided by wave functions as being analogous to car trajectories guided by roads. If the roads are ordered, then so are the car trajectories. If the roads are messy, then the car trajectories are messy too.

The wavefunction in Bohmian mechanics is said to be unknowable. In a 3D (single-particle) case, we can determine the wavefunction using the Schrodinger's equation and the boundary conditions we in principle know from the geometry. However, can we ever know complete universal 3N-D wavefunction in the configuration space?

I mean, the wavefunction in Bohmian mechanics is kinda postulate, in all papers I have seen, you HAVE it along with the initial configuration, and then you determine the future behavior of the system. In other words, the universal wavefunction is not something fixed and it could be something else (It can be said to be "random"). So how come can it be well-shaped? I mean, I think that well-shaped trajectories in pictures are just simplified cases assuming one single particle going toward the two slits. I'd like to know your opinion.

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