Adiabatic Spin Tuning

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.