In a Letter published in the January 2021 edition of this journal, B. H. Suits described coming across an online site where an erroneous version of the $R 3 , 1 ( r )$ hydrogenic radial wavefunction appeared.1 The formula was a factor of 3 too large; Suits encouraged posters of such expressions to test them in advance by running a quick numerical integration to check the normalization.

Curious as to how widespread errors involving hydrogenic radial wavefunctions might be in traditional texts, I surveyed 15 introductory quantum mechanics texts.2–16 This selection includes ones I had on hand plus a few volumes I consulted at a library or which were readily accessible online and which did not reference any other source for their formulae. All list radial wavefunctions or overall wavefunctions from which the radial part can be extracted for principal quantum numbers n ranging from 2 to 6; publication dates spanned several decades. To check the normalization of these expressions, I developed a spreadsheet into which they could be programmed in order to run numerical integrations. All but one were correct, with the lone exception being the $R 3 , 1 ( r )$ function in Dicke and Wittke being too large by a factor of 3; this was probably the source of the error Suits encountered.3

A related question is whether these functions are orthogonal; that is, what can be said of
$⟨ R n 1 , ℓ 1 , R n 2 , ℓ 2 ⟩ = ∫ 0 ∞ R n 1 , ℓ 1 ( r ) R n 2 , ℓ 2 ( r ) r 2 d r .$
It is, in fact, easy to show that some products of dissimilar R's will not be orthogonal. In general, the form of these functions is $R n , ℓ ( r ) ∼ r ℓ L n − ℓ − 1 2 ℓ + 1 ( r ) e − r / n a o$, where $L n − ℓ − 1 2 ℓ + 1 ( r )$ denotes an associated Laguerre function, which is a polynomial in r with terms of orders $r 0 , r 1 , … , r n − ℓ − 1$. If $ℓ = n − 1$, the overall form reduces to $R n , ℓ ∼ r n − 1 e − r / n a o$, so we must obviously have $⟨ R n 1 , n 1 − 1 , R n 2 , n 2 − 1 ⟩ ≠ 0$. As for other possibilities, I have run numerical integrations on all combinations up to n = 5; it quickly became clear that orthogonality holds when identical $ℓ$-states are involved: $⟨ R n 1 , ℓ , R n 2 , ℓ ⟩ = δ n 1 n 2$.

This question does not appear to be addressed in any detail in standard texts. The complication is that the conventional orthogonality relationship for associated Laguerre polynomials involves an argument which must be the same for the two functions being multiplied, but in the case of the hydrogenic wavefunctions, this argument depends on the principal quantum number involved; the relationship does not apply if the two n's are different. This issue turns up in an exercise in Arfken's text on mathematical physics; the orthogonality relation can be found in Arfken (Eq. (13.48)) and Eyring et al. (Sec. 4h).17,18 This behaviour must be intrinsic to the associated Laguerre polynomials; I would be interested in hearing from readers who might know of a proof and corresponding physical interpretation.

Nowadays, it is easy to plot special functions with dedicated software, a situation which makes it tempting to skip over details of their analytic forms and behaviors when in class. However, these functions form the basis of our understanding of electron orbitals and molecular bonding; taking time to explore them is still worthwhile.

The author declares no conflict of interest.

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Avoid propagation of typos with numerical methods
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