From Bohm to Classical Trajectories in a Hydrogen Atom

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.
To show how Bohmian trajectories (circular orbits) smoothly transform to classical trajectories (elliptical orbits), a stationary state with the quantum numbers , and is considered. The motion of the particle is inextricably linked with its environment through the quantum potential . The quantum potential does not depend on the intensity of the wave, but only on its functional form. It need not decrease with increasing distance. The environment coupling function is chosen such that the system behaves fully classically in the limit and fully quantum mechanically if . For every , the orbit ends up in a stable classical cycle as time increases. The starting point of the first trajectory (green) is . Near the nodal point , the trajectories begin to rotate about the axis very quickly, which leads to an unstable motion in the mesoscopic case.
In the graphic you see the wave density (if enabled); two possible orbits, where one trajectory (cyan) depends on the initial starting point (, , ); and the initial starting points of the two trajectories (black points, shown as small spheres).


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Since the Coulomb potential is spherically symmetric, the Schrödinger equation can be solved analytically in spherical polar coordinates, with equal to the distance between the nucleus and the electron, derivatives written , and so on:
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.
[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."
[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)
[5] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Apr 14, 2017)
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