# From Bohm to Classical Trajectories in a Hydrogen Atom

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.

n=2

l=1

m=1

Q

b

b≫0

b=0

b>0

(3,3,-3)

(0,0,0)

z

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).

x

0

y

0

z

0