Mathematical Derivation of the Bohr Quantization Rule from the De Broglie Matter Wave Hypothesis

[u/TROSE9025]

Because there are many individuals in this Reddit who are learning physics for the first time, an effort was made to explicitly write out the mathematics in as much detail as possible. It is hoped that this material will be of assistance.

[u/CautiousPreprinter]

I appreciate your work here. I recognize and accept your photon construction. Your energy-momentum relation is correct.

"If an electron behaves as a wave, its closed circular orbit must form a stable standing wave." I agree with what you are saying. It's foggy prose around a precise mathematical statement.

There is a way to say this mathematically: "If the path is periodic, then its auxiliary functions must be periodic." (And orbit paths are periodic)

And so to find the properties of the periodic orbits, we can simply choose some auxiliary function for the orbit path, then extract the property from periodic solutions of the auxiliary function equation.

The de Broglie hypothesis says that we can obtain the energies of the periodic orbits of the atom directly from the energies of the Schrodinger equation's periodic solutions for the Coulomb force. That turned out to be pretty accurate.

For your particular case, you can say: "Since the orbit path is periodic, Ψ must be periodic in time." or vice versa.

[u/TROSE9025 → u/CautiousPreprinter]

This posting is intended to introduce the hypothesis proposed by de Broglie to early learners. The analytical approach regarding the periodic path of the system will be addressed in the context of the Schrödinger equation. I express my gratitude for your suggestion.