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As we know, the principal quantum number $ n=1,2,3,... $ is related to the radial quantum number $ n_r=0,1,2,... $ by $$ n=n_r+\ell+1 .$$ What is the physical (or chemical) definition of the radial quantum number $n_r$?

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To complement @EmilioPisanty's answer: the importance of $n_r$ and the number of nodes is that, for fixed $L$, the energy increases with $n_r$. The restriction to fixed $L$ comes because, for a given potential, the energy increases with $n_r$. Here, the potential is the effective potential which contains the centrifugal barrier $\sim \ell(\ell+1)/r^2$ terms. The increase with $n_r$ is a general feature of the solutions of the Schrodinger equation, and the centrifugal term is present in every spherically-symmetric potential.

Since $n=n_r+\ell+1$, it is clear that increasing $n_r$ increases $n$ and thus increases the energy. Examples below illustrate the probability densities as a function of $r$ for $\ell=0$ and $\ell=2$ states, and also show that increasing $n_r$ correspond to increasing $n$ and thus to increasing energy. Note how the scale of the horizontal axis (the $r$ axis) changes with $n$ because the tail of the hydrogen solutions are of the form $e^{-r/(2n a_0)}$ so that larger $n$ solutions extend well beyond smaller $n$ solutions.

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The radial quantum number counts the number of radial nodes in the wavefunction:

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