I'm struggeling to solve this question. Can anyone help me please?
Let us consider a generic quantum mechanical system governed by the Hamiltonian $\ H(t) $.
In what follows we denote the evolution operator by $\ u(t, t_0) $. Hence, $\ |Ψ(t)> = u(t, t_0)|Ψ_0> $ satisfies the time-dependent Schrodinger equation, where $\ |Ψ_0> $ represents the wave function at $\ t = t_0$.
(a) Prove the unitarity condition $\ u^† (t, t_0)u(t, t_0) = u(t, t_0)u^†(t, t0) = I $ .
(b) Now let us assume that the system under study exhibits a symmetry represented by an antilinear (and antiunitary) operator $U$ which has nothing to do with the time reversal. Show that in this case $\ [u(t, t0), U] = 0 $, and thus the system is unstable since its spectrum is unbounded from below.
(c) Show that instability disappears if antilinear $U$ includes time reversal.
The conclusion from (b) and (c) is that symmetries represented by antilinear operators are possible, but they necessarily involve time reversal.