In class 25

25.1

$$|{\mu }_0|=|g|\frac{{\mu }_B^e}{\hslash }|S|=|g|\frac{2}{2m_e}=2\frac{e}{2m_e}|S|$$ $$|{\mu }_p|=|g|\frac{{\mu }_B^p}{\hslash }|S|=|g|\frac{2}{2m_p}=5.6\frac{e}{2m_p}|S|$$ $$\frac{|{\mu }_e|}{|{\mu }_p|}=\frac{2m_p}{5.6m_e}=\frac{2}{5.6}\frac{1.67E-27}{9.09E-31}=660$$

25.2

For spin up

$$\cos \theta =\frac{\frac{1}{2}\hslash }{\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)\hslash }}=\frac{1}{\sqrt{3}}⟹\theta =54.7^{\circ }$$

For spin down

$$\cos \theta =\frac{-\frac{1}{2}\hslash }{\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)\hslash }}=\frac{-1}{\sqrt{3}}⟹\theta =125.3^{\circ }$$

25.3

$$g=-2$$ $$\hat{H}=-{\hat{\mu }}_e\cdot B=-({\hat{\mu }}_e{)}_z{B}_z=-\frac{ge{B}_z}{2m_0}{\hat{S}}_z={\omega }_0{\hat{S}}_z$$

Eigenvalues:

$$+\frac{\hslash {\omega }_0}{2},-\frac{\hslash {\omega }_0}{2}$$ $$\Delta E=\hslash {\omega }_0=\frac{2e{B}_z\hslash }{2m_0}=\frac{2\times 1.6E-19{\times }_1\times \hslash }{2\times 9.1E-31}$$ $${\omega }_0=1.76E11$$ $$\lambda =\frac{2mc}{{\omega }_0}=\frac{2\pi \times 3E8}{1.76E11}=1.1cm$$

25.4

Prove $[{\hat{S}}_x,{\hat{S}}_y]=i\hslash {\hat{S}}_z$

$$[{\hat{S}}_x,{\hat{S}}_y]={\hat{S}}_x{\hat{S}}_y-{\hat{S}}_y{\hat{S}}_x$$ $$=\frac{\hslash }{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\frac{\hslash }{2}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)-\frac{\hslash }{2}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\frac{\hslash }{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$ $$-\frac{{\hslash }^2}{2}\left(\begin{array}{cc}0& i\\ -i& 0\end{array}\right)=i\hslash {\hat{S}}_z$$