The three-dimensional stationary Schrödinger equation with Coulomb potential
can be written in parabolic coordinates
with the wavefunction
, the Laplace operator
, the partial derivative
with respect to time and the Coulomb potential
. For simplicity, we set atomic units
In spherical polar coordinates, the unnormalized energy eigenfunctions for the hydrogen atom associated with the quantum state with the quantum numbers
is given by:
is a spherical harmonic. The radial function is
is a generalized Laguerre polynomial.
For a stationary case with
, the phase function
reduces to zero. Thus the Bohm momentum
of the ground state electron is
, and finally the electron is at rest . For
, the electron is in motion.
If the wavefunction is expressed in parabolic coordinates (for more details see [4–6]), we find:
is a Whittaker function.
The energy spectrum depends only on the integers
For the anisotropic Coulomb potential, the Schrödinger equation in Cartesian coordinates is:
Due to computational limitations, the superposed total wavefunctions
are taken as a sum of just three eigenfunctions
, where the unnormalized wavefunction
for a particle is written as:
Each part of the superposition of the three stationary state has the same energy:
The parabolic wavefunction
could be expressed by a linear combination of stationary states of spherical wavefunction
. For example,
for convenience. After variable transformations from parabolic coordinates to Cartesian coordinates, we obtain the total phase function
The gradient of the phase
is obtained from the total wavefunction in polar form,
, with the quantum amplitude
. The velocity field
is calculated by the Bohm momentum
. From the Bohm momentum
it follows that
In the case of the superposition of the three stationary states with equal energy, the corresponding differential equation system for the velocity field becomes autonomous and obeys the time-independent part of the continuity equation
with the wave density
in parabolic coordinates reduces to
with the total phase function
This could be simplified for
The trajectories reduce to circles with the velocities:
The velocity depends on the position of the particle only. For
, most of the orbits form "spirally circles" along the
axis, in which a spiral orbit is perpendicular on the circle orbit (for an example, see ).
In all other cases, the integration of the velocity vector
in respect to time leads to complex periodic or chaotic orbits (for examples, see [8, 9]). If
), it is strongly implied that chaotic motion occurs.
are increased (if enabled), the results will be more accurate.
 P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics
, New York: Cambridge University Press, 1993.
 L. I. Schiff, Quantum Mechanics
, 2nd ed., New York: McGraw-Hill, 1955.
 E. Merzbacher, Quantum Mechanics
, New York: Wiley, 1961.
 J. F. Ogilvie, "The Hydrogen Atom According to Wave Mechanics: Part II. Paraboloidal Coordinates," Revista de Ciencia y Tecnologia
(2), 2016 pp. 25–39. arXiv:1612.05098