I'm looking for the link between the Rydberg formula for hydrogen spetcral series
$$\frac{1}{\lambda_{\mathrm{vac}}} = R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$
and this image.
Is it right to say that the Balmer Series is created by all transitions from (x, _, _) to (2, _, _) where x>2 and _ are "don't cares"?
Are the three (2, _, _) states all of the same energy?
The 'link' is, in short, the Schrödinger equation.
The orbitals plotted in the image ─ the wavefunctions $\psi_{n,l,m}(r,\theta,\phi)$ ─ are the hydrogenic solutions to the hydrogen Schrödinger equation, $$ \left[ -\frac{\hbar^2}{2m}\nabla^2 - \frac{e^2}{4\pi\epsilon_0} \frac{1}{r} \right]\psi_{n,l,m}(r,\theta,\phi) = E_{n,l,m} \:\psi_{n,l,m}(r,\theta,\phi), $$ for which you require the energy to be $$ E_{n,l,m} = E_{n} = -\frac{m e^{4}}{(4\pi\epsilon_0)^{2}\hbar ^{2}}{\frac {1}{2n^{2}}}, $$ independently of $l$ and $m$.
Thus,
Are the three (2, _, _) states all of the same energy?
Yes.
Is it right to say that the Balmer Series is created by all transitions from (x, _, _) to (2, _, _) where x>2 and _ are "don't cares"?
Yes, that is correct (though it's important to note that selection rules generally apply, and not all the transitions within that set actually contribute any significant signal).
For further details, see any introductory textbook on quantum mechanics.
