The Rotating Wave in de Broglie-Bohm Mechanics

A free Gaussian wave packet solution of the Schrödinger equation can be converted to a wave packet rotating about the direction of motion. This Demonstration considers a three-dimensional Gaussian wave packet in the de Broglie–Bohm approach (often called Bohmian mechanics). In this theory, the particle has a well-defined trajectory in configuration space calculated from the total phase function. In practice, it is impossible to predict or control the quantum trajectories with complete precision. Real three-dimensional space is taken as the configuration space in this context.
In the de Broglie–Bohm approach, the possible orbits and velocities for this special wavefunction depend on the initial density and initial position of the starting particles in the three-dimensional configuration space. The particle rotates faster around the axis if the starting point is closer to the origin.
The graphics show the wave density (if enabled), the initial starting points of 32 possible orbits (shown as small yellow spheres), the actual positions (shown as small colored spheres) and 32 possible trajectories with the initial distance .
  • Contributed by: Klaus von Bloh
  • In memory of Detlef Dürr, who passed away on January 3, 2021.

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DETAILS

A Gaussian wave packet solution obeys the free-particle Schrödinger equation in cylindrical coordinates (with derivatives written and so on):
.
This could be converted to Cartesian coordinates (with atomic units: ):
.
This Demonstration uses a three-dimensional Gaussian wave packet solution in atomic units, which is slightly different from [1]:
with , the initial width and the wave number in the direction.
Wave packets that rotate about their direction of motion could be constructed using derivatives with respect to :
.
If is independent of , or in Cartesian coordinates generally:
,
which has an angular momentum component along the direction.
In this special case, the wave packet is given by:
.
It follows that in Cartesian coordinates:
where .
In Cartesian coordinates, the components of the velocity can be determined from the gradient of the total phase function, which leads to:
,
,
,
where the component of the velocity is given by , with in the trajectory.
The magnitude of the resultant velocity vector is given by
,
or, in cylindrical coordinates:
.
References
[1] J. Lekner,"Rotating Wavepackets," European Journal of Physics, 29(5), 2008 pp. 1121–1125. doi:10.1088/0143-0807/29/5/025.
[2] K. von Bloh, The Rotating Wave in the de Broglie Bohm Approach (Bohmian Mechanics) [Video]. (Jul 28, 2021) www.youtube.com/watch?v=u_Cux59xiWA.
[3] Bohmian-Mechanics.net. (Jul 24, 2021) bohmian-mechanics.net.
[4] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Jul 24, 2021) plato.stanford.edu/entries/qm-bohm.
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