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.