In Example 8.3

, we introduced a simple quantum system involving an electrically charged, spin- \(1 / 2\) particle immersed in a uniform magnetic field. In this exercise, we will continue the study that was started there, and calculate more expectation values and establish uncertainty principles for the particle's spin.

(a) What is the uncertainty principle for energy and the \(x\)-component of the spin of the electron? That is, for variances \(\sigma_{E}^{2}\) and \(\sigma_{S_{x}}^{2}\) for the energy and \(x\)-component of the spin, respectively, what is their minimum product:

\[\begin{equation*}\sigma_{E}^{2} \sigma_{S_{x}}^{2} \geq ? \tag{8.150}\end{equation*}\]

(b) When does the lower bound in the uncertainty relation vanish? Show that this makes sense from when we know the variances vanish.