The sp Hybrid Orbital in Bohmian Mechanics

Hybrid orbitals are linear combinations of degenerate (equal energy) hydrogen atom orbitals of different angular momentum [1, 2]. The orbitals are spherical, while the orbitals have lobes that can point in different directions.
In this Demonstration, the -hybridization of a hydrogen atom with a constant phase shift is investigated in the Bohmian mechanics description. In this case, the orbital is combined with one of the orbitals to yield two hybrid orbitals [1], but here only one lobe of the orbital is shown. These hybridized orbitals result in higher electron density, depending on the phase shift , in the bonding region for a bond. The hybrid orbital is plotted in three dimensions. In the Bohm picture [3, 4], the electron acts like an actual particle, its velocity at any instant being fully determined by the gradient of the phase function, and here the constant phase shift influences only the magnitude of the resultant velocity vector but not the shape of the orbits.
In the graphic, you see the wave density (if enabled); the magnitude of the resultant velocity vector (if enabled); nine possible orbits of one electron, where the trajectories (colored) depend on the initial starting point (, , ), the initial starting points of the trajectories (black points, shown as small spheres) and the actual position (colored points, shown as small spheres).

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For hybrid orbitals, the wavefunction is represented as a linear combination of degenerate eigenfunctions of the one-electron hydrogen atom, with principal quantum number , angular momentum quantum number and magnetic quantum number .
The one-electron Schrödinger equation for the Coulomb potential can be solved analytically in spherical polar coordinates.
For simplicity, set the reduced mass , the Bohr radius and equal to 1 (atomic units). This leads to the time-dependent wavefunction with the associated Laguerre polynomials and the spherical harmonics :
.
A degenerate superposition of two eigenstates for only one lobe of the orbital with a constant phase shift in the normalized form gives the total wavefunction:
or in Cartesian coordinates
.
For a constant phase shift with , the velocity becomes an autonomous differential equation system. The influence of a constant phase shift is described in [5].
From the total wavefunction for , the equation for the phase function follows:
.
For with , the total phase function becomes independent of the variables , and (); therefore the velocity equals zero.
In Cartesian coordinates, the components of the velocity could be determined by the gradient of the total phase function from the total wavefunction in the eikonal form (often called polar form), which leads in this special case to a corresponding autonomous differential equation system:
with
,
,
and
.
The magnitude of the resultant velocity vector is defined as:
.
For with , the constant phase shift influences only the magnitude of the resultant velocity vector and not the form of the orbits.
References
[1] R. E. Dickerson and I. Geis, Chemistry, Matter, and the Universe: An Integrated Approach to General Chemistry, Menlo Park, CA: W. A. Benjamin, 1976.
[2] Wikipedia. "Orbital Hybridisation." (Jun 26, 2019) en.wikipedia.org/wiki/Orbital_hybridisation.
[3] "Bohmian-Mechanics.net." (Jun 26, 2019) www.bohmian-mechanics.net/index.html.
[4] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jun 26, 2019) plato.stanford.edu/entries/qm-bohm.
[5] K. von Bloh. "Influence of the Relative Phase in the de Broglie-Bohm Theory" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/InfluenceOfTheRelativePhaseInTheDeBroglieBohmTheory.
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