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I have been reading Griffith's Introduction to Quantum Mechanics, and I just went over the solution of the infinite spherical well. He gives it as $$\psi _{nlm}(r,\theta, \phi) = A_{nl}j_l\left(\beta_{Nl} \frac{r}{a}\right) Y^{m}_l (\theta, \phi)$$ Where $n, l, m$ are the principal quantum numbers. $l, m$ are the eigenvalues that come out of solving the angular portion of the spherical PDE.

I don't quite understand the presence of both $n, N$ in the solution. This is how he gets to them: \begin{align} u''(r)&=\left(\frac{l(l+1)}{r^2}-k^2\right)u, k\equiv\frac{\sqrt{2mE}}{\hbar} \\ \implies u(r) &= Arj_l(kr)+Brn_l(kr)\\ \end{align} Since, } u(r)&=R(r)/r \begin{align} R(r) &= Aj_l(kr)+Bn_l(kr)\\ \end{align} Since $R$ is finite at $r=0$, \begin{align} R(r)&=Aj_l(kr)\\ BC: R(a)&=0\\ k&=\frac{1}{a}\beta _{Nl} \end{align} where $\beta _{Nl}$ is the $N$th zero of $j_l$. So, $$E_{Nl} = \frac{\hbar ^2}{2ma^2} \beta ^2 _{Nl}$$ And then claims that the radial wave function $$ \psi _{nlm}(r,\theta, \phi) = A_{nl}j_l\left(\beta_{Nl} \frac{r}{a}\right) Y^{m}_l (\theta, \phi)$$ has $N-1$ radial nodes. My question is, how do $n$, $N$ relate to each other here? Why does the radial wave function have $N-1$ nodes, if $j_l$ has an infinite number of zeros?

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There is a footnote in that edition that explains this apparent 'typo', which actually is not. Griffiths claims that $N$ is related to $n$ and $\mathcal{l}$, but in a quite complicated way for the infinite spherical well case (that's why he prefer to distinguish between $N$ and $n$). He also mentions that for the particular case of the H atom there is a really simple relation between those indices: $n = N + l$. This might sound confusing but it is just that $n$ orders energy levels while $N$ has to do with radial nodes, as shown in Fig. 4.3.

Hope it helps.

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In my edition of Griffiths (2nd US edition), there is no $N$ in these equations; all of your $N$'s are $n$'s in my edition. Maybe your edition has a typo? If you're using a later edition, I would speculate that they switched from $n$ to $N$ in the indices in the third edition to avoid confusion with the spherical Bessel functions, but didn't quite make all the corrections they should have.

As far as your question about $j_l$: By definition, the argument of $j_l(\beta_{Nl} r/a)$ is greater than $\beta_{Nl}$ for $r > a$, which is outside the region of interest. So the zeros beyond the $N$th zero are not relevant, since the wavefunction is zero for $r > a$. Only the $N-1$ zeros that occur when the argument of $j_l$ is smaller than $\beta_{Nl}$ (i.e., when $r < a$) are physically meaningful here.

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  • $\begingroup$ Do they denote spherical Neumann functions as $n$? How unfortunate and confusing, especially since usually cylindrical Neumann functions are denoted as $Y$ and thus spherical ones as $y$... $\endgroup$
    – Ruslan
    Commented Jun 16, 2021 at 21:44
  • $\begingroup$ Isn't $jl$ supposed to be $J_l$? $\endgroup$
    – Gert
    Commented Jun 16, 2021 at 22:47
  • $\begingroup$ @Gert: I don't believe so, no. The lower case $j$ typically denotes spherical Bessel functions while the upper case $J$ typically denotes cylindrical Bessel functions. The two are related, of course. $\endgroup$ Commented Jun 17, 2021 at 0:14
  • $\begingroup$ Ah yes, that explains it. Thanks. $\endgroup$
    – Gert
    Commented Jun 17, 2021 at 17:23
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As you have said $$j_{l}(0)=\beta_{N,l}$$ has infinite zeros for a given l.

The first zero belongs to n=1, the second for n=2 the 3rd to n=3, and soon http://www.sc.ehu.es/sbweb/fisica3/cuantica/separacion/esferica.html . $$\hspace{1in}$$Allways l=N-1 as Bohr model.

As soon as, $$l<N$$ the first zero, for example, $$\beta_{1,1}$$ has not physical meaning, but exists mathematically. https://www.researchgate.net/publication/374925994_About_Cavity_Potentials <DOI: 10.13140/RG.2.2.19627.62247>

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