Details

A Gaussian wave packet solution obeys the free-particle Schrödinger equation in cylindrical coordinates (with derivatives written ∂θu=∂ ∂ θu and so on):

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This could be converted to Cartesian coordinates (with atomic units: ):

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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 :

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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:

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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:

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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.