Time reversal and motion reversal

[hokhani]

Does time reversal operation changes the time "t" to time" -t"( For example if we are at t=10 s then by applying time reversal operator the time turns into t=-10 s?), or time reversal operation operates instantly in such a way that if it operates on a ket at t=10 s it only reverses the motion at exactly t=10 s?

 

[DrClaude]

Could you please specify what you mean by "time reversal operator"?

What is usually meant by time reversal is that if you have a system in state $| \psi (t_1)\rangle$ at time $t_1$ and evolve it to time $t_2 > t_1$ and find $| \psi (t_2)\rangle$, then you will find that you can recover state $| \psi (t_1)\rangle$ by evolving state $| \psi (t_2)\rangle$ "backward in time", by simply changing $t$ to $-t$ in the Schrödinger equation. In other words, the TDSE is symmetric with respect to time reversal.

 

[hokhani]

Could you please specify what you mean by "time reversal operator"?

Time reversal operator is an anti unitary operator $\Theta$ and $\Theta \psi (x,t)=\psi^*(x,-t)$. Your statement is $U \Theta \psi(t_2)= \Theta \psi (t_1)$ in which $U$ is time evolution operator and I guess that $\Theta \psi (x,t)=\psi^*(x,-t)$ means instant change of time in reverse direction and not changing t to -t.

 

[DrDu]

No, the time reversal operator acts on all t.