Magnetism and Electrons — Question 3

Question 3

Electron spin projections differ from the spin-moment magnitude

Use electron spin s=1/2, $m_s=\pm1/2$, $g_s=2$, and $\boldsymbol\mu_s=-g_s\mu_B\mathbf S/\hbar$. The spin magnitude is $\hbar\sqrt{s(s+1)}$.

1. Find the spin-moment magnitude and allowed z projections.

2. For a measurement $S_z=+\hbar/2$ in a positive z field B, find the magnetic energy and identify whether it is the lower or upper level.

Question 3 - Solution

Electron spin projections differ from the spin-moment magnitude

(a) Method and calculation

$$|\boldsymbol\mu_s|=g_s\mu_B\sqrt{s(s+1)}=\boxed{\sqrt3\mu_B}.$$$$\boxed{\mu_{sz}=-g_s\mu_Bm_s=\mp\mu_B}.$$

(b) Interpretation and further calculation

$$S_z=+\hbar/2\Rightarrow\mu_{sz}=-\mu_B\Rightarrow U=-\mu_{sz}B=+\mu_BB.$$$$\boxed{U\text{ is the upper level};\quad U_{\mathrm{lower}}=-\mu_BB}.$$

Physical check

Spin is intrinsic angular momentum; it is not the mechanical rotation of a charged sphere.