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1

Only the mathematical formulas are time reversible. That does not mean that the law itself is reversible. You can understand it this way - Every law/formula has an implicit condition that says - "time flows only forward". Other way to understand can be that when the coffee cup falls, universe (gravity in this case) makes the cup fall. There is no force/law ...

3

It is not really correct to call them " reversible laws of physics" Laws of physics lead to mathematical models that describe observations. These models are usually differential equations of space and time. The solutions of these equations exist both for time going towards infinity as for time going towards minus infinity in most cases, for mechanics, ...

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The standard reply is: get your video camera out, take a movie of dropping a (empty) coffee cup on the kitchen floor, then play the movie backwards. There is no physical way of telling which way time is running from your movie or the equations behind the event (Newtons Laws). Rather than thinking of an absolute arrow of time, so the coffee cup will break ...

2

Yes, the gaussian wavepacket can get narrower as the time passes indeed. It's a matter of phases. You know that a gaussian wavepacket is the superposition of plane waves, each one having a precise wavevector. So it really depends on how you "prepare" this superposition, i.e. on how you set the phase of each chromatic component. If at $t=0$ all the plane ...

2

There is a mistake in your definition of time reversal as $x$ is fixed under that transformation, the remaining transformations being correct. With this correct version of T, the Hamiltonian you study is PT symmetric.

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There is yet another solution (maybe more elementary)$^1$, with some components of the answers from Qmechanic and JoshPhysics (Currently I'm taking my first QM course and I don't quite understand the solution of Qmechanic, and this answer complement JoshPhysics's answer) the solution uses the Heisenberg Equation: The time evolution of an operator $\hat{A}$ ...

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