# Effect of time reversed hamiltonian acting on a state?

• A
• Brage
In summary, the conversation discusses the effect of a time reversed Hamiltonian on a state ket, where the Hamiltonian is not time-invariant. The conversation goes on to explain the thought process and potential error in the assumption. The participants also clarify the use of the time reversal operator and the role of linearity in the 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 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 can't seem to find it.

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

Brage

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
Jilang said:
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