# Why do you use $n_r = n -\ell -1$ as quantum number instead of $n$ for hydrogen atom?

I got two different quantum numbers for the same problem: Hydrogen atom without any interaction.

Then, my energy is $$E_n = -\frac{R_y}{n^2}$$ with the quantum number $$n = 1, 2, 3, ....$$

In another version it is $$E_{n_r} = -\frac{R_y}{(n_r+\ell+1)^2}$$ with the quantum number $$n_r = n - \ell -1$$.

Unfortunately, there is no explanation why it uses $$n_r$$ in the second formula and what the difference physically is between $$n$$ and $$n_r$$. I guess the index stands for the quantum number due to the radial part of the wave function $$R_{n\ell}$$ but I am confused because it works totally fine with $$n$$.

Is there any explanation written somewhere or can anybody explain?

• Commented Aug 25, 2022 at 15:39
• – Yodo
Commented May 9, 2023 at 10:07

The Hamiltonian of the hydrogen atom is a central potential $$H_{\rm H}\propto\frac{1}{r}$$ this means that the angular-momentum is conserved. This translates to quantum mechanics as a decomposition in terms of spherical harmonics and thus you only need to quantum numbers $$n_r$$ and $$l$$ to describe the energies. However that is not all, the hydrogen atom has a hidden symmetry, that is the conservation of the Runge-Lenz vector, which translates into a spectrum that only needs one quantum number $$n$$. The link between symmetries and conservations is given by Noether's theorem.

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 $$\ell=0$$.

There is no difference. You can use either formula.

The radial quantum number $$n_r$$ is equal to the number of nodes in the radial part of the wave-function $$\psi$$ (the spheres where $$\psi(\textbf r) = 0$$, excluding the origin and infinity).

The principal quantum number $$n$$ is often defined as

$$n := n_r + \ell + 1$$

and it represents the energy levels of the electron in the gross-structure approximation.

Because an atom is three dimensional and the nucleus-electron interation is spherically symmetric - thus the resulting potential is spherically symmetric. The wavefunction can have angular components in addition to radial components in that case.

All this means is that in addition to $$n$$, you also get $$l$$ as a quantum number together defines the energy state of an electron. If you review from page 4 of this link: http://www.damtp.cam.ac.uk/user/tong/qm/qm4.pdf you may get a better picture of what I'm talking about.