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  1. For classical systems, are time reversal invariant (T.R.I) and time-independent (T.I) necessarily the same thing?

  2. For quantum systems, are time reversal invariant (T.R.I) and time-independent (T.I) the same thing?

Can one give examples for classical and quantum that are:

(1) T.R.I and T.I. systems

(2) T.R.I but not T.I. systems

(3) T.I but not T.R.I systems

(4) not T.R.I and not T.I. systems, but unitary.

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They are absolutely not the same thing. There are many time reversal invariant systems in classical mechanics, for example a planet orbiting a star, however nothing will ever be time independent. Imagine that you thought you had a system which is time independent, it consists of a bunch of things and none of them are moving. Relativity will tell you that you can move at a constant velocity and physics will be the same. In which case it will appear that everything in the system is moving uniformly and is not time independent.

Quantum mechanically, it's essentially the same, it is easy to come up with time reversal invariant systems, for example the quantum harmonic oscillator. And there are no time invariant systems in quantum mechanics because relativity still holds and you could use the previous argument.

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    $\begingroup$ I am suspecting that you are confused with time-reversal symmetry transformation $T$ (the discrete transformations not within the continuous Lorentz group) and the Lorentz transformation $\Lambda_{1/2}$ (the 6 generators). They are not the same operations. (p.s. I did not vote in case someone willl vote later.) $\endgroup$
    – wonderich
    Dec 28, 2017 at 19:47

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