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This is obviously not specific enough. For example, kinetic energy is frame dependent, so you always have to provide the reference...

Thanks, but as I read in some references, internal energy is equal to the sum of the average kinetic and potential energies of all...

Suppose that we just put a cylinder of an ideal gas in a higher place. Does it's internal energy increase?

Try Fetter&Walecka or Coleman.

Look at this notes from Berkeley: https://bohr.physics.berkeley.edu/classes/221/1112/notes/timerev.pdf They do the usual time-reversal...

The notes you sent formalizes the time reversal in a nice approach in accord with Sakurai. The many body book of Bruus is a good...

I think if we accept the complex conjugation only acts on the numbers there is no ambiguity in this term, because at least the explicit...

Eq. (7.18) in the text, introduces the time reversal dependent Hamiltonian as: ##H\psi(r)=[-\frac{1}{2m}(\nabla_r+ieA)^2+V(r)]\psi(r)##...

Right, since in that text the Hamiltonian were represented in position space where ##H_r## is diagonal, I implicitlly assumed diagonal...

Thanks, it was the key point here. If we work in the basis ##|q\rangle## where ##\Theta |q\rangle =|q\rangle##, then we have: ##\Theta H...

Sorry this gets confusing because we mix various stuff in second quantization, your ##H## is in some basis, let me use hats for...

From the Eq. (7.18) in that text, it seems that by ##H## the authors mean the Hamiltonian in the position space. Also, as I remember, it...

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts...

Thanks: right, your method was very nice, but I would like to try this method which still I am stuck in that.

In general, if the operator ##C = A B## is made out of two operators ##A## and ##B##, then from $$ \langle \psi, C \psi \rangle =...