# Understanding the solution of the infinite spherical well

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?

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.

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.

• 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 $y$... Commented Jun 16, 2021 at 21:44
• Isn't $jl$ supposed to be $J_l$?
– Gert
Commented Jun 16, 2021 at 22:47
• @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. Commented Jun 17, 2021 at 0:14
• Ah yes, that explains it. Thanks.
– Gert
Commented Jun 17, 2021 at 17:23

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, $$l the first zero, for example, $$\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>