# Effect of time reversed hamiltonian acting on a state?

• A
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 throught 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 cant seem to find it.

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

Brage

## Answers and Replies

How does the first equality in your third paragraph arise?

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

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!

• Jilang