Quantum Contradictions 3: Hydrogen Orbitals

The electron structure of an atom is supposed to be described by a wave function defined by 4 (integer) quantum numbers (n,l,m,s) with  n = 1,2,..., the principal (shell) quantum number,  l = 0,.., n -1, the azimutal quantum number, m ranging from -l to +l the magnetic quantum number  and s = +- 1/2 the spin quantum number.

The 3 quantum numbers (n,l,m) describe the wave functions for excited states (eigen-functions or orbitals) of the Hydrogen atom, with the 1 wave function for n = 1 representing the s-state of the first shell, the 1+ 3 = 4 wave functions for n = 2 representing the p-states of the second shell, and the 4 + 5 = 9 wave functions for n = 3 representing the d-states of the the third shell et cet. The connection is that the Hydrogen eigen-functions form a basis allowing the electronic structure of a multi-electron atom to be represented by linear combinations.

We see that the number of Hydrogen orbitals equals n^2, which is to be compared with the observed number 2n^2 electrons in a complete shell, with He (2 electrons in shell 1), Ne (2 electrons in shell 1 and 8 in shell 2), Ar (2+8+8), Kr (2+8+18+8), et cet. A factor 2 is thus missing and this factor was introduced by adding a 4th quantum number as the two-valued spin quantum number s. 

The spin quantum number thus came out as a forced resolution of a contradiction between observed electronic shell structure with 2n^2 electrons in a complete shell with a structure represented by n^2 Hydrogen orbitals. But a motivation that a multi-electron shell structure should resemble the s-p-d structure of the orbitals of the one-electron Hydrogen atom, was missing.

The structure of the periodic table is thus claimed to be inherited from the structure of Hydrogen orbitals augmented by spin, as the basic experimental support of quantum mechanics.  But there are irregularities of the periodic table which require additional (ad hoc) assumptions such as Madelung's and Hund's rules, and so the question to be considered in an upcoming post is:

• Does standard quantum mechanics  explain the periodic table?.