Adiabatic Spin Tuning

Classical Mechanics Level 3

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.

\frac{5\hbar}{\mu B}

\frac{\hbar}{2\mu B}

\frac{2\hbar}{\mu B}

\frac{\hbar}{10 \mu B}

1 solution

Matt DeCross
May 10, 2016

This problem is an application of the energy-time uncertainty principle.

If the electron is expected to remain in the spin-up ground state after the process ends, the probability of the electron jumping the energy gap throughout the process should stay negligible. This is true as long as the time to complete the process is long enough that the energy uncertainty stays smaller than the energy gap.

The minimum energy gap occurs when the B field is at strength $\frac{B}{10}$ ; the gap is $2\frac12 \mu\frac{B}{10} = \mu \frac{B}{10}$ from the given formula for the energy. Plugging into the energy-time uncertainty principle, the minimum time is:

$\sigma_t \geq \frac{\hbar}{2\sigma_E} = \frac{\hbar}{2} \frac{10}{\mu B} = \frac{5\hbar}{\mu B},$

as claimed.

Adiabatic Spin Tuning

1 solution

0 pending reports