Think an electronic structure of an atom having Z electrons.

Naturally the solution of the simple Shroedinger equation for the electronic state, i.e., the wave function becomes a 3Z-variable function. Mathemathecally speaking, the problem is identical to an eigen value problem for a 3Z-dimensional-one-body system! The would-be ground state solution will be symmetrical with respect to all these 3Z variables. That is, all of the electron will be described by the same mathematecal representation as if they are all staying in one same state.

The problematic point is this.

The modern view on the electronic structure of an atom demands the one-body approximation and description using such concepts like K-shell, L-shell, or orbital angular momentum quantum number, or magnetic angular momentun quatum number, etc. That is, electrons are occupying each spin-orbital that are energitically different with each other, respectively.

The simple solution of the simple Schroedinger equation will never give such an intricate answer. Then, I would like to ask, what is wrong in the above argument?

What kind of operator or operators should be added further to the original Hamiltonian in order to derive the wave function of the modern elecgtronic structure, mathematecally? What is your opinion, Sirs?

Thanks for reading

If memory serves, the reason is that electrons are fermions, and the Pauli exclusion principle states that no two fermions may occupy the same quantum state (i.e. have the same quantum numbers). The electronic structure of a hydrogen-like atom is given by the solution to the Schrödinger equation, the full derivation of which I will defer to someone better versed in physical chemistry. The quantum numbers arise from the radial and angular (polar and azimuth) terms in the solution. Additional operators are not necessary.