# From Bohm to Classical Trajectories in a Hydrogen Atom

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A continuous transition between quantum Bohm trajectories and classical motion is demonstrated. This might be of interest in applications to mesoscopic systems. The hydrogen atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. The Schrödinger equation in spherical polar coordinates for the hydrogen atom is solved by separation of variables. The solutions can exist only when certain constants that arise in the solution are restricted to integer values.

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Contributed by: Partha Ghose and Klaus von Bloh (April 2017)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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For simplicity, set the reduced mass and equal to 1 (atomic units). This leads to the time-dependent wavefunction with the associated Laguerre polynomials and the spherical harmonics :

,

with the energy eigenvalue and . The quantum number determines the energy, is called the *orbital quantum number*, and the order is the *magnetic quantum number*. For , the energy equals . For , , , in Cartesian coordinates with , and , the wavefunction becomes:

.

From the wavefunction for , the equation for the phase function follows:

,

and therefore for the components of the velocity:

, and .

To see how the Bohmian trajectories smoothly pass over to classical trajectories, the general equation of motion is given by the acceleration term from classical mechanics, which is the second derivative of the position with respect to time :

.

Here is the quantum potential, is the Coulomb potential with , is the environment coupling function, and is defined by:

with and .

For our special case, the quantum potential becomes:

For more detailed information about the continuous transition between classical and Bohm trajectories, see [1–3] and for an animated example, see [4]. For more detailed information about Bohmian mechanics, see [5].

The results become more accurate if you increase AccuracyGoal, PrecisionGoal and MaxSteps.

References

[1] P. Ghose, "A Continuous Transition between Quantum and Classical Mechanics. I," *Foundations of Physics*, 32(6), 2002 pp. 871–892. doi:10.1023/A:1016055128428.

[2] P. Ghose and M. K. Samal, *"*A Continuous Transition between Quantum and Classical Mechanics. II,"* Foundations of Physics*, 32(6), 2002 pp. 893–906. doi:10.1023/A:1016007212498.

[3] P. Ghose and K. von Bloh, "Continuous Transitions between Quantum and Classical Motions." arxiv.org/abs/1608.07963.

[4] P. Ghose and K. von Bloh, *Continuous Transitions Between Quantum and Classical Motion for Three Electrons in the Hydrogen Atom* [Video]. (Apr 14, 2017) www.youtube.com/watch?v=9pryk-I5Ki0.

[5] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Apr 14, 2017)plato.stanford.edu/entries/qm-bohm.

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