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:

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:

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.

[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.