Critique on various ways to think about time reversal transformation on Schrodinger equation?
Please define how time-reversal symmetry act on <PERSON> equation $i \frac{\partial}{\partial t} |\Psi{}(t) \rangle = H(t) |\Psi{}(t) \rangle.$ (for general form: which can be relativistic such as Dirac theory or nonrelativistic.)
Below I present various ways to think about time reversal transformation on Schrodinger equation.
Could you present your critique or criticism on which one make sense? What else is the correct way to do time reversal transformation on <PERSON> equation?
1.
The <PERSON> equation governs the quantum system (regardless of relativistic or not) says: $$ i \frac{\partial}{\partial t} |\Psi{}(t) \rangle = H(t) |\Psi{}(t) \rangle. $$ It is easy to check the <PERSON> equation is time-reversal $T$ invariant if
(1). $H(t)$ has no explicit time-dependent, or time-reversal symmetric, then $ H(-t)=H(t)=H$.
(2). 669
Since we do not want to make $t=0$ becomes a special fixed point, we also require the time translational symmetry $H(t)=H(t+\Delta)$ for some arbitrary time interval $\Delta$.
(3) Under time interval transformation, if we think from the perspective of assign the value of a statevctor $|\Psi{}(-t) \rangle$ (or its projected wavefunction) at $-t$ to a new statevector $|\Psi'{}( t) \rangle$ at $t$ following the classical mechanics approach advocated by <PERSON> https://physics.stackexchange.com/a/633205/42982, we get $$ |\Psi{}( t) \rangle \overset{T}{\to} |\Psi'{}( t) \rangle = |\Psi{}(-t) \rangle. \tag{1} $$
We can do a time-reversal transformation on the whole equation acting on the state vector (a ray) in the Hilbert space, $$ T i \frac{\partial}{\partial t} |\Psi{}(t) \rangle = T H(t) |\Psi{}(t) \rangle $$ $$ =T i T^{-1} T\frac{\partial}{\partial t} |\Psi{}(t) \rangle = T H(t) T^{-1} T|\Psi{}(t) \rangle $$ $$ =T i T^{-1} T\lim_{\Delta \to 0}\frac{ |\Psi{}(t+\Delta) \rangle- |\Psi{}(t) \rangle}{\Delta} = H(-t) |\Psi{}(-t) \rangle $$ Then we adopt and plug in eq.(1): $$=Ti T^{-1} \lim_{\Delta \to 0}\frac{ |\Psi{}(-t-\Delta) \rangle- |\Psi{}(-t) \rangle}{\Delta} = H(-t) |\Psi{}(-t) \rangle $$ $$ =T i T^{-1} (-)\frac{\partial}{\partial \tilde t} |\Psi{}(\tilde t) \rangle \Big \vert_{\tilde t = -t} = H(-t) |\Psi{}(-t) \rangle. $$
As I show, the <PERSON> equation is time-reversal $T$ invariant requiring that (1) $H(-t)=H(t)$ and (2) $$T i T^{-1} =-i$$ which is the case that the time-reversal is an antiuntiary and antilinear symmetry.
The above has a problem because we do only map $ |\Psi{}( t) \rangle \overset{T}{\to} |\Psi'{}( t) \rangle = |\Psi{}(-t) \rangle $ but do not really do a complex conjugation on the (complex valued) wavefunction. 804