# 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.

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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

## Snapshots

## 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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