I went over a calculation of the hydrogen wavefunction using Dirac equation (this one) and I am a bit confused by the angular part. The final result for the wavefunction based on that derivation is this:

$$ \begin{pmatrix} if(r) Y_{j l_A}^{m_j} \\ -g(r) \frac{\vec{\sigma}\cdot\vec{x}}{r}Y_{j l_A}^{m_j} \end{pmatrix} $$

where $f(r)$ and $g(r)$ are radial functions and $Y_{j l_A}^{m_j}$ are spin spherical harmonics. In the derivation they show that $Y_{j l_A}^{m_j}$ and $-\frac{\vec{\sigma}\cdot\vec{x}}{r}Y_{j l_A}^{m_j}$ differ in their value of orbital angular momentum, $l$ by 1 and they have opposite parities. For example, if $j=1/2$, $Y_{j l_A}^{m_j}$ can have $l=1$ and $-\frac{\vec{\sigma}\cdot\vec{x}}{r}Y_{j l_A}^{m_j}$ would have $l=0$ (or the other way around). This implies (as it is mentioned in that derivation) that $l$ ($L^2$ as an operator) is not a good quantum number for a Dirac spinor.

I am not sure how to think about this. For example the atomic states are usually labeled as $^{2S+1}L_{J}$, which implies that the state has a definite orbital angular momentum, l. Is that just an approximation? Another thing I don't understand is the parity. As we are dealing only with electromagnetism, the wavefunctions should have a definite parity. But the top and bottom part in the spinor above have opposite parities, so it looks like the Dirac spinor doesn't have a defined parity. Can someone explain to me how should I think about these spinors? Should I look only at the top part? I know the bottom part is ignored in non-relativistic limit, but parity should still be a good quantum number even in the relativistic case (where I can't just ignore the bottom part).

Thank you!



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