Abstract
The phase velocity  of a material wave is given by the following equation:  If this formula is rewritten using the Planck-Einstein relation  and the de Broglie formula , it becomes:  Next, the values for E and p, obtained from a relation applicable to the electron in a hydrogen atom derived by the author, are substituted into this equation. When that is done, multiple formulas relating to the phase velocity of the electron wave are derived. These formulas contain still-unknown ultra-low energy levels of the hydrogen atom and electron orbital radii  corresponding to those energy levels.  also appear in the formula for energy levels of the hydrogen atom derived in this paper. This serves as grounds for the existence of ultra-low energy levels in addition to the already-known energy levels of the hydrogen atom.
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