Origin of $ell leq n-1$ orbital rule

Question

I am wondering about the origin of the $ell leq n-1$ orbital filling rule. For the hydrogen atom, I believe the reason is because in the spatial wave function there is the term
$$psi propto sqrt{(n-ell-1)!}$$
so if $ell > n-1$, by the definition of the factorial, $psi$ goes to $0$. However, what about atoms with more protons and electrons? Will we always have this sort of term in our wavefunction or is there are more general argument for why this inequality has to be true?

Qmechanic · Answer

TL;DR: The quantization condition
$$n_r ~:=~ n-ell -1 ~inmathbb{N}_0 tag{A}$$
follows by looking for normalizable wavefunction solutions $R(r)$ to the radial TISE.
In more detail: Schematically, one first solves $R(r)$ in the regions for small & large radial coordinate $r$. After factoring out the newly found asymptotic behaviours, one obtains a function $v(r)$, where (due to the TISE) the coefficients of its power series satisfy a recursion relation. It turns out that the series $v(r)$ must truncate in order for the solution $R(r)$ not to alter its asymptotic behaviour. This leads to the quantization condition (A), cf. e.g. Ref. 1.
References:

D. Griffiths, Intro to QM, 1995; subsection 4.2.1.

Origin of $ell leq n-1$ orbital rule

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