The energy of an electron in a magnetic field $$B$$ is, in some unit system:

$E = \pm \frac{1}{2} \mu B,$

where choice of positive or negative sign corresponds to spin-down or spin-up respectively, and $$\mu$$ is the spin magnetic moment of the electron.

In an adiabatic transition, the parameters of a quantum system are gradually changed to bring a system smoothly from one state to another state. Suppose an electron starts in the spin-up ground state in a magnetic field of strength $$B$$. The magnetic field is then reduced slowly to strength $$\frac{B}{10}$$ and then increased slowly again back to strength $$B$$. Find the minimum time for the process of tuning the magnetic field to occur for which the electron is expected to remain in the spin-up ground state after the process ends. Hint: consider the energy-time uncertainty principle.

Note: this is a very simple demonstration of the fact that adiabatically tuning electron spins requires relatively long time scales.

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