Effect of time reversed hamiltonian acting on a state?

[Brage]

Hi, I have been trying to get my head around the effect of a time reversed hamiltonian $H^B(t)=H(-t)=T^{-1}H^F T$ on a state ket $|\psi>$, where $H^F=H$ is the regular hamiltonian for the system (energy associated with forward time translation) and $H^B=H(-t)$ is the time reversed hamiltonian, and $T$ is the time reveral operator. I here assume the hamiltionian is not time-invariant. Let me explain my through process:

As $i\partial_t = H$ this implies that $T^{-1}i\partial_t T=T^{-1}H^F T=H^B$.

But $T^{-1}i\partial_t T=-T^{-1}iT\partial_t=i\partial_t$, as $T^{-1}iT = -i$. Which would seem to imply that $H^F|\psi>=H^B|\psi>$, which seemingly contradicts the assumed condition $[H^F,H^B]\neq 0$. I assume this means I have made a mistake somewhere but can't seem to find it.

I would appreciate any help from people who can point out my error, cheers!

Brage

 

[Jilang]

How does the first equality in your third paragraph arise?

 

[Brage]

Well $T\partial_t |\psi>=\partial_{-t}T|\psi>$ so then $T\partial_t |\psi>=-\partial_{t}T|\psi>$ correct?

 

[Jilang]

I agree with the first equality, but I don't see how the second equality follows unless the state is linear in t.

 

[Brage]

I agree with the first equality, but I don't see how the second equality follows unless the state is linear in t.

Oh of course I was using $d(-t)=-dt$. Cheers for that!