The energy of an electron in a magnetic field is, in some unit system:
where choice of positive or negative sign corresponds to spin-down or spin-up respectively, and 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 . The magnetic field is then reduced slowly to strength and then increased slowly again back to strength . 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.