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ZeroTheHero
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The radial quantum number $n_r$ is related to the number of nodes in the radial part of the wavefunction since the associated Laguerre polynomial $$ R_{n+\ell}^{2\ell+1}(r/a_0) $$$$ R_{n_r}^{2\ell+1}(r/a_0) $$ is a polynomial of degree $n_r=n-\ell-1$. Moreover, since $n\ge 1$, this also gives a bound on the largest possible $\ell$ for a given $n$ since obviously $n_r\ge 0$. Thus, if $n=1$ then it must be that the largest (and only possible value of) $\ell$ is $\ell=0$.

The radial quantum number $n_r$ is related to the number of nodes in the radial part of the wavefunction since the associated Laguerre polynomial $$ R_{n+\ell}^{2\ell+1}(r/a_0) $$ is a polynomial of degree $n_r=n-\ell-1$. Moreover, since $n\ge 1$, this also gives a bound on the largest possible $\ell$ for a given $n$ since obviously $n_r\ge 0$. Thus, if $n=1$ then it must be that the largest (and only possible value of) $\ell$ is $\ell=0$.

The radial quantum number $n_r$ is related to the number of nodes in the radial part of the wavefunction since the associated Laguerre polynomial $$ R_{n_r}^{2\ell+1}(r/a_0) $$ is a polynomial of degree $n_r=n-\ell-1$. Moreover, since $n\ge 1$, this also gives a bound on the largest possible $\ell$ for a given $n$ since obviously $n_r\ge 0$. Thus, if $n=1$ then it must be that the largest (and only possible value of) $\ell$ is $\ell=0$.

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ZeroTheHero
  • 47.8k
  • 21
  • 68
  • 147

The radial quantum number $n_r$ is related to the number of nodes in the radial part of the wavefunction since the associated Laguerre polynomial $$ R_{n+\ell}^{2\ell+1}(r/a_0) $$ is a polynomial of degree $n_r=n-\ell-1$. Moreover, since $n\ge 1$, this also gives a bound on the largest possible $\ell$ for a given $n$ since obviously $n_r\ge 0$. Thus, if $n=1$ then it must be that the largest (and only possible value of) $\ell$ is $\ell=0$.