- #1

Malamala

- 308

- 27

$$

\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!