# Time reversal of a QM Hamiltonian

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 $$t \rightarrow -t$$.
• Thanks! So, looks like it changes the sign under both $P$ and $T$, and is, therefore, $PT$-symmetric. – mavzolej Feb 7 at 17:41