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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?

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  • $\begingroup$ 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. $\endgroup$ – rob Dec 13 '20 at 18:29
  • $\begingroup$ @rob no because the spherical harmonics are not swapped. $\endgroup$ – Jeanbaptiste Roux Dec 13 '20 at 18:39
  • $\begingroup$ Using that "k" is an unnecesary complication. $\endgroup$ – DanielC Dec 14 '20 at 19:17
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I am not sure the formula is correct. The expression under the square root $\sqrt{\pm 2 k+1}$ seems negative.

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  • $\begingroup$ It seems like you would choose the negative sign only to make the expression well-behaved for negative $k$. $\endgroup$ – rob Dec 13 '20 at 18:27
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    $\begingroup$ @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:-) $\endgroup$ – akhmeteli Dec 13 '20 at 18:42
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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 $k<0$.

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