Effects of Hamiltonian Preferred Basis on Decoherence

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What would be the effects on the system for different values of the Hamiltonian preferred basis in Decoherence? Would it for example make the electrons higher in orbital or bands? Or what would be the exact effects?
 
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What do you mean by "preferred basis"? You choose a basis by choosing a measurement device, because then you measure the corresponding observable, and that's why you evaluate the corresponding probabilities from the given state via Born's rule. That's all what's behind the term "a preferred basis", one sometimes read in texts on "interpretation".
 
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