# Spin spherical functions in hydrogen-like solution to Dirac equation

I saw a video where the following equality is wrote for the angular part of the solution of the Dirac equation for hydrogen-like atoms: \begin{align} \Omega^{j=l\pm 1/2}_{lm}&= \frac{1}{\sqrt{2l+1}}\left( \begin{array}{c} \displaystyle \pm \sqrt{l\pm m+\frac{1}{2}}\,Y_{l,m-1/2}(\theta,\varphi)\\ \displaystyle \sqrt{l\mp m+\frac{1}{2}}\,Y_{l,m+1/2}(\theta,\varphi) \end{array} \right)\\&= \frac{1}{\sqrt{\pm 2k+1}}\left( \begin{array}{c} \displaystyle \sqrt{\pm k- m+\frac{1}{2}}\,Y_{\pm k,m-1/2}(\theta,\varphi)\\ \displaystyle -\text{sgn}(k)\sqrt{\mp k +m+\frac{1}{2}}\,Y_{\pm k,m+1/2}(\theta,\varphi) \end{array} \right) \end{align} Where $$k=-l-1$$ for $$j=l+1/2$$ and $$l$$ for $$j=l-1/2$$, and $$Y_{l,m}$$ are the spherical harmonics. I don't know how to prove this equality, especialy the $$\text{sgn}(k)$$ term. I can't figure out why the $$\pm m$$ and $$\mp m$$ are changed into $$-m$$ and $$+m$$ and why the $$\pm$$ sign disapears in the first line. Anyone knows how to prove this?

• Is the order of the two lines swapped? That is, the strictly positive expression is second in the $\ell$ formulation, but first in the $k$ formulation. – rob Dec 13 '20 at 18:29
• @rob no because the spherical harmonics are not swapped. – Jeanbaptiste Roux Dec 13 '20 at 18:39
• Using that "k" is an unnecesary complication. – DanielC Dec 14 '20 at 19:17

I am not sure the formula is correct. The expression under the square root $$\sqrt{\pm 2 k+1}$$ seems negative.
• It seems like you would choose the negative sign only to make the expression well-behaved for negative $k$. – rob Dec 13 '20 at 18:27
• @rob : I would think one has to choose the same sign as in the "left-most" hand side (containing $\Omega$). Anyway, I am skeptical about a formula copied from a video:-) – akhmeteli Dec 13 '20 at 18:42
The spherical harmonics $$Y_{\ell m}$$ are nonzero for $$0 \leq |m| \leq \ell$$ and are polynomials in $$\cos\theta$$ of order $$\ell$$. (Note that, when tabulated, they are frequently simplified using the identity $$\cos^2\theta - 1 = \sin^2\theta$$.)
The notation $$Y_{\pm k, m+1/2}$$ suggests that we must choose the sign so that $$k$$ is positive. However, you've said the options are $$k=\ell$$ or $$k=-\ell-1$$. There is no recurrence relation where $$Y_{\ell,m}$$ is a polynomial with the same order as $$Y_{|-\ell-1|,m}$$, so the two sets of expressions are different if $$k<0$$.