How Does n^2 Equal the Max Number of Electron Orbitals in Quantum Mechanics?

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kilo_Z71
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Does anyone know how to prove, mathmatically, that the principle quantum number squared equals the maximum number of electon orbitals?any help would be appricated!

n^2=max # electron orbitals
 
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This might be for proving number of electroncs in each orbital not the max orbitals but i think its similar. Its from the Schrödinger equation. Basically he looked at the fact that each electron has a wavefunction and a wavelength and that only intergral numbers of electroncs (thus wavelengths) could exist in each orbital (cant have half an electron) so given the distance from the nucleus and thus the circumferance you can work out how many wavelengths fit in.

I think
-G