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Can we proof the relation that the principal quantum number $n$ and azimuthal quantum number $l$ have the relation $l=0,1,...n-1$ in any spherical symmetric potential $V(r)$ or this just apply to Coulomb potential?

I would appreciate reference to books or articles

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This relationship does not hold for all spherically-symmetric potentials. For example, for the 3D harmonic oscillator the relationship between $n$ and $l$ is

$$l=0,2,4,...,n$$

for even $n$ and

$$l=1,3,5,...,n$$

for odd $n$.

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  • $\begingroup$ So why in multi electrons atoms we have this relation,since in multi electron atoms we approximate the potential to be spherically-symmetric potentials? $\endgroup$
    – amilton moreira
    Dec 25, 2018 at 18:59
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    $\begingroup$ @amiltonmoreira But we do more than this. To first approximation (the orbital approximation) we consider each electron to have hydrogen-like orbitals created from an effective nuclear charge $Z_{eff}$ created from the nuclear charge and shielding of inner electrons. See chemicool.com/definition/orbital-approximation.html $\endgroup$
    – isometry
    Dec 25, 2018 at 20:29
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The radial $P(r,\nu,l)$ part of the wave function in a central potential obey a one dimensional Schrodinger equation an so it is characterized by the number of nodes $\nu$, and since the equation contains $l$ it is also characterized by the quantum number $l$. Now we define $n=l+\nu+1$

So instead of characterizing the radial part by $l$ and $\nu$ we characterize it by $l$ and $n$ ,that is, the radial part of the wave function is of the form $P(r,n,l)$. Now since the number of nodes $\nu$ and angular number $l$ are positive integer we should have $l=0,1,...n-1$

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