I suggest, where is your place? The purpose is that you must either give a reference or a thing equivalent,Should you be expressing that uncertianty theory is presently taken under consideration.
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The problem here is that people are convinced the HUP is usually a starting position in QM in lieu of simply a consequence. It truly is as if a person needs to forcibly introduce the HUP into something, as opposed to realizing that it is naturally inbuilt when a single adopts the QM formalism.
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Consequently finite current and vanishing entropy are not contradictory, even though QM effects are taken into consideration.
It seems the kind of reference you are immediately after would have to become a general evidence with the uncertainty basic principle, since the 'i's label quantum states, which necessarily satisfy the HUP. I'm certain most QM textbooks will provide a derivation. Consider Sakurai, Fashionable Quantum Mechanics second edition for example.
classical calculation will not implement as I already stated.It can be pointed out in feynman lectures vol. one that uncertainty theory has to here be invoked for non zero entropy.see some early chapter,it's written there.