I'm interested in the time reversal properties of a term in the non-relativistic QM Hamiltonian proportional (up to a true scalar) to $$ H \propto (\vec S_1 \times \vec S_2) \cdot \vec L $$ The situation with $\vec L$ is clear, it does change the sign. What about the first term in the product? Doesn't its parity depend on the particular spin state?


$\mathbf{L}$, $\mathbf{J}$ and $\mathbf{S}$ all change sign under time reversal.

For $\mathbf{L}$ this is trivial, since it depends on $\mathbf{p} = \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}$ and with $t \rightarrow -t$ you get $\mathbf{p} \rightarrow -\mathbf{p} $.

For spin, I guess at the end of the day it just worked. But the procedure is rooted in treating spin as a magnetisation $\propto \mathbf{M}$.
And $\mathbf{M}$, like $\mathbf{B}$, are generated by currents so by terms like $\frac{\mathrm{d}q}{\mathrm{d}t}$, so again $t \rightarrow -t$.

See here for more discussion.

  • $\begingroup$ Thanks! So, looks like it changes the sign under both $P$ and $T$, and is, therefore, $PT$-symmetric. $\endgroup$ – mavzolej Feb 7 at 17:41
  • $\begingroup$ Well technically it should obey CPT symmetry. I guess it does because coulomb interactions don’t care about spin? So it’s always gonna be C symmetric? $\endgroup$ – SuperCiocia Feb 7 at 17:44

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