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For free electrons in a metal, the Fermi energy $E_F$ (defined as the highest filled energy level at $T=0$). It represents a characteristic energy scale of the system as well as the energy of the electrons at the Fermi level. What is the significance of the natural time scale $\hbar E^{-1}_F$?

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The Fermi energy is best thought as a level, rather than some absolute number: that is, if you shift the zero of energy up or down, the energy of the Fermi level will change accordingly.

In that regard, it is similar to the difference between the ionization potential of an atom and its ground-state energy: the ionization potential is the minimal energy difference between the ground-state energy and a continuum state, so it will not change if you re-set your energy zero to some other value (like e.g. to the start of some higher ionization continuum, say, the $\mathrm{He}^{2+}+2e^-$ doubly ionized state of helium, which would allow for states with an unbound single electron with negative total energy), while the ground-state energy does change in such a re-setting of conventions.

Within that paradigm, the Fermi energy is like the ground-state energy as opposed to being like the ionization potential. As such, it does not have an unambiguous physical meaning as an energy difference, which is what you would need for the time $\hbar/E_F$ to be physically meaningful.

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Maybe this value limits the time of the existence of the state from below, because we can't have the uncertainty in energy more than Fermi energy

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