I Dirac equation in the hydrogen atom

Malamala
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Hello! 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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The spinor above is indeed a valid solution of the Dirac equation and so it does have a definite parity. However, the parity of the two components of the spinor are opposite, which is why the angular part of the wavefunction does not have a definite orbital angular momentum. In the non-relativistic limit, the bottom component is ignored, so the wavefunction has a definite parity and a definite angular momentum. In terms of labeling atomic states, it is common to use the approximate non-relativistic labels, even when considering relativistic effects. This is because in many cases, the relativistic corrections are small and so the non-relativistic labels are still good approximations.
 
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