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TL;DR: The quantization condition $$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 \tag{A}$$ follows by looking for normalizable wavefunction solutions $R(r)$ to the radial TISE.

In more detail: Schematically, one first solves $R(r)$ in the regions for small & large radial coordinate $r$. After factoring out the newly found asymptotic behaviours, one obtains a function $v(r)$, where (due to the TISE) the coefficients of its power series satisfy a recursion relation. It turns out that the series $v(r)$ must truncate in order for the solution $R(r)$ not to alter its asymptotic behaviour. This leads to the quantization condition (A), cf. e.g. Ref. 1.

References:

1. D. Griffiths, Intro to QM, 1995; subsection 4.2.1.

TL;DR: The quantization condition $$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 \tag{A}$$ follows by looking for normalizable wavefunction solutions $R(r)$ to the radial TISE with a fixed $\ell$ value. Here the radial quantum number $n_r$ is the number of nodes, cf. the node theorem in 1D QM.

In more detail: Schematically, one first solves $R(r)$ in the regions for small & large radial coordinate $r$. After factoring out the newly found asymptotic behaviours, one obtains a function $v(r)$, where (due to the TISE) the coefficients of its power series satisfy a recursion relation. It turns out that the series $v(r)$ must truncate in order for the solution $R(r)$ not to alter its asymptotic behaviour. This leads to the quantization condition (A), cf. e.g. Ref. 1. $\Box$

Classically, $n_r=0$ corresponds to circular orbits, i.e. no radial motion.

References:

1. D. Griffiths, Intro to QM, 1995; subsection 4.2.1.

TL;DR: The quantization condition $$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 \tag{A}$$ follows by looking for normalizable wavefunction solutions $R(r)$ to the radial TISE.

In more detail: Schematically, one first solves the $R(r)$ in the regions for small & large radial coordinate $r$. After factoring out the newly found asymptotic behaviours, one obtain a function $v(r)$, where the coefficients of its power series satisfy a recursion relation. It turns out that the series must truncate in order for the solution not to alter its asymptotic behaviour. This leads to the quantization condition (A), cf. e.g. Ref. 1.

References:

1. D. Griffiths, Intro to QM, 1995; subsection 4.2.1.

TL;DR: The quantization condition $$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 \tag{A}$$ follows by looking for normalizable wavefunction solutions $R(r)$ to the radial TISE.

In more detail: Schematically, one first solves $R(r)$ in the regions for small & large radial coordinate $r$. After factoring out the newly found asymptotic behaviours, one obtains a function $v(r)$, where (due to the TISE) the coefficients of its power series satisfy a recursion relation. It turns out that the series $v(r)$ must truncate in order for the solution $R(r)$ not to alter its asymptotic behaviour. This leads to the quantization condition (A), cf. e.g. Ref. 1.

References:

1. D. Griffiths, Intro to QM, 1995; subsection 4.2.1.

The quantization condition $$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 $$ follows by looking for normalizable wavefunction solutions to the radial TISE. See e.g. Ref. 1 for details.

References:

1. D. Griffiths, Intro to QM, 1995; subsection 4.2.1.

TL;DR: The quantization condition $$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 \tag{A}$$ follows by looking for normalizable wavefunction solutions $R(r)$ to the radial TISE.

In more detail: Schematically, one first solves the $R(r)$ in the regions for small & large radial coordinate $r$. After factoring out the newly found asymptotic behaviours, one obtain a function $v(r)$, where the coefficients of its power series satisfy a recursion relation. It turns out that the series must truncate in order for the solution not to alter its asymptotic behaviour. This leads to the quantization condition (A), cf. e.g. Ref. 1.

References:

1. D. Griffiths, Intro to QM, 1995; subsection 4.2.1.