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
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
, the velocity becomes an autonomous differential equation system. The influence of a constant phase shift is described in .
From the total wavefunction for
, the equation for the phase function
, the total phase function
becomes independent of the variables
); 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:
, the constant phase shift
influences only the magnitude of the resultant velocity vector and not the form of the orbits.
 R. E. Dickerson and I. Geis, Chemistry, Matter, and the Universe: An Integrated Approach to General Chemistry,
Menlo Park, CA: W. A. Benjamin,