In class 26

In class 26.1

$${\omega }_0=-\frac{5.6\ast 21\ast 1.602E-19}{2(1.67E-27)}=5.6\times 10^9$$ $$f=\frac{{\omega }_0}{2\pi }\approx 900Mz$$

In class 26.2

a.

$$\theta =\frac{\pi }{2},\varphi =0,\psi =\frac{A}{\sqrt{2}}({\chi }_{+}+{\chi }_{-})$$

Normalized:

$$\frac{A^2}{2}+\frac{A^2}{2}=1,A=1$$ $$\psi =\frac{1}{\sqrt{2}}({\chi }_{+}+{\chi }_{-})$$

b.

$$\frac{\hslash }{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)=\frac{\hslash }{2}\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$$

Eigenvalue:

$$\frac{\hslash }{2}$$

26.3

$$\Delta S_z=\sqrt{<s_z^2>-<S_Z{>}^2}$$ $${\hat{S}}_z=\frac{\hslash }{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$ $${\hat{S}}^2=\frac{{\hslash }^2}{4}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$ $$\begin{gathered} <{\hat{S}}_Z>=\frac{1}{\sqrt{2}}(1 \\ 1)\frac{{\hslash }^2}{4}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)=\frac{{\hslash }^2}{4} \end{gathered}$$ $$\Delta S_z=\frac{\hslash }{2}$$

26.4

Dr. Wertz said there is a 0 probability