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:

,

,

.

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.

[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.