How Does n^2 Equal the Max Number of Electron Orbitals in Quantum Mechanics? (2024)

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

This might be for proving number of electroncs in each orbital not the max orbitals but i think its similar. Its from the Schrodinger 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

is a fundamental concept in quantum mechanics and can be mathematically proven using the Schrödinger equation. The Schrödinger equation is a mathematical equation that describes the behavior of particles at the quantum level, including electrons in an atom.

In the Schrödinger equation, the principle quantum number (n) represents the energy level or shell that the electron occupies, while the maximum number of electron orbitals is determined by the angular momentum quantum number (l). The maximum number of electron orbitals is given by the formula 2l+1.

Using the Schrödinger equation, we can derive the energy levels for a hydrogen atom. The energy levels are given by the formula E=-13.6/n^2 eV, where n is the principle quantum number.

Now, if we substitute the maximum number of electron orbitals formula (2l+1) for n, we get E=-13.6/(2l+1)^2 eV. We can then simplify this equation to E=-13.6/l^2 eV.

We know that the energy levels for a hydrogen atom are discrete and that the energy of an electron in an orbital cannot be between two energy levels. This means that the energy level must be a whole number, and therefore the principle quantum number must also be a whole number.

If we square the principle quantum number (n^2), we get n^2=l^2, which is the same as the maximum number of electron orbitals (2l+1). This shows that the principle quantum number squared is equal to the maximum number of electron orbitals.

In conclusion, the principle quantum number squared equals the maximum number of electron orbitals because it is derived from the Schrödinger equation and is consistent with the discrete energy levels of a hydrogen atom. This mathematical proof supports the fundamental concept of quantum mechanics and helps us understand the behavior of electrons in an atom.