The Rotating Wave in de Broglie-Bohm Mechanics

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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.
Contributed by: Klaus von Bloh (August 2022)
In memory of Detlef Dürr, who passed away on January 3, 2021.
Open content licensed under CC BY-NC-SA
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Details
A Gaussian wave packet solution obeys the free-particle Schrödinger equation in cylindrical coordinates (with derivatives written ∂θu=∂ ∂ θu 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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