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Eigenvalues are a unique set of scalar values associated with a set of linear equations, most commonly found in matrix equations.

Characteristic roots are another name for eigenvectors. It's a non-zero vector that can only be altered by its scalar factor once linear transformations are applied.

In quantum physics, Eigen spinors are considered basis vectors that represent a particle's general spin state.

Use eigenvalues $|S_r-位I|=0$and find the value the eigenvalues $S_r$,$位$.

$$\begin{gathered} \left|\begin{array}{cc}\frac{魔}{2}\cos 胃-位& \frac{魔}{2}{e}^{-i蠒}\sin 胃\\ \frac{魔}{2}{e}^{-i蠒}\sin 胃& \frac{魔}{2}\cos 胃-位\end{array}\right|=0 \\ -\frac{魔}{4}{\cos }^2胃+{位}^2-\frac{魔}{4}{\sin }^2胃=0 \\ {位}^2=\frac{魔}{4}\left({\sin }^2胃+{\cos }^2胃\right) \\ {位}^2=\frac{-魔}{4} \\ 位=卤\frac{魔}{2} \\ \mathrm{Thus},\mathrm{the}\mathrm{eigenvalues}\mathrm{of}{\mathrm{S}}_{\mathrm{r}}\mathrm{is}卤\frac{\mathrm{魔}}{2} \end{gathered}$$

Use the eigen-spinor as an example$\left(\begin{array}{c}a\\ b\end{array}\right)$.

$S_2\left(\begin{array}{c}a\\ b\end{array}\right)=位\left(\begin{array}{c}a\\ b\end{array}\right)$

For $位=\frac{魔}{2}$,

localid="1659013250803" $$\begin{gathered} \frac{魔}{2}\left(\begin{array}{cc}\cos 胃& e^{-i蠒}\sin 胃\\ e^{-i蠒}\sin 胃& -\cos 胃\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\frac{魔}{2}\right) \\ \left(\begin{array}{cc}\cos 胃& e^{-i蠒}\sin 胃\\ e^{-i蠒}\sin 胃& -\cos 胃\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}a\\ b\end{array}\right) \end{gathered}$$

Compare the relevant entries on both sides of the matrices.

$$\begin{gathered} a\cos 胃+b{e}^{-i蠒}\sin 胃=a \\ a{e}^{i蠒}\sin 胃-b\cos 胃=b \end{gathered}$$

Solve the above two equations.

$$\begin{gathered} b=\frac{a\left(1-\cos 胃\right)}{e^{-i蠒}\sin 胃} \\ =e^{-i蠒}\frac{\left(1-\cos 胃\right)a}{\sin 胃} \\ =e^{-i蠒}\frac{\left(2{\sin }^2{\displaystyle \frac{胃}{2}}\right)}{2{\sin }^2{\displaystyle \frac{胃}{2}}\cos {\displaystyle \frac{胃}{2}}}a \\ =e^{-i蠒}\frac{\sin {\displaystyle \frac{胃}{2}}}{\cos {\displaystyle \frac{胃}{2}}}a \end{gathered}$$

For $\left(\begin{array}{c}a\\ b\end{array}\right)$become commonplace,

Apply ${\left|a\right|}^2+{\left|b\right|}^2=1$.

${\left|a\right|}^2+\frac{{\sin }^2\left({\displaystyle \frac{胃}{2}}\right)}{{\cos }^2\left(\frac{胃}{2}\right)}{\left|a\right|}^2=1$

Solve the above expression further.

$$\begin{gathered} {\left|a\right|}^2\left[\frac{{\sin }^2\left({\displaystyle \frac{胃}{2}}\right)}{{\cos }^2\left({\displaystyle \frac{胃}{2}}\right)}+1\right]=1 \\ {\left|a\right|}^2\left[\frac{\sin \left({\displaystyle \frac{胃}{2}}\right)+{\cos }^2\left({\displaystyle \frac{胃}{2}}\right)}{{\cos }^2\left(\frac{胃}{2}\right)}\right]=1 \\ {\left|a\right|}^2\left[\frac{1}{\cos {\displaystyle \frac{胃}{2}}}\right]=1 \\ a=\cos \frac{胃}{2} \end{gathered}$$

Substitute the above value in $b=e^{i蠒}\frac{\sin {\displaystyle \frac{胃}{2}}}{\cos {\displaystyle \frac{胃}{2}}}a$.

$$\begin{gathered} b=e^{i蠒}\frac{\sin {\displaystyle \frac{胃}{2}}}{\cos {\displaystyle \frac{胃}{2}}}\cos \frac{胃}{2} \\ =e^{i蠒}\sin \frac{胃}{2} \end{gathered}$$

Substitute the values of a, and b in $\left(\begin{array}{c}a\\ b\end{array}\right)$.

$\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}\cos \frac{胃}{2}\\ e^{i蠒}\sin \frac{胃}{2}\end{array}\right)$

For $位=\frac{-魔}{2}$,

$$\begin{gathered} \frac{魔}{2}\left(\begin{array}{cc}\cos 胃& e^{-i蠒}\sin 胃\\ e^{-i蠒}\sin 胃& -\cos 胃\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=-\frac{魔}{2}\left(\begin{array}{c}a\\ b\end{array}\right) \\ a\cos 胃+b{e}^{-i蠒}\sin 胃=-a \\ a{e}^{i蠒}\sin 胃-b\cos 胃=b \end{gathered}$$

Solve the above two equations.

$$\begin{gathered} b=\frac{-a\left(1+\cos 胃\right)}{\sin 胃}{e}^{i蠒} \\ =-a{e}^{i蠒}\frac{2{\cos }^2{\displaystyle \frac{胃}{2}}}{2\sin {\displaystyle \frac{胃}{2}}\cos {\displaystyle \frac{胃}{2}}} \\ =-a{e}^{i蠒}\frac{2{\cos }^2{\displaystyle \frac{胃}{2}}}{2\sin {\displaystyle \frac{胃}{2}}\cos {\displaystyle \frac{胃}{2}}} \\ \mathrm{Apply}{\left|\mathrm{a}\right|}^2+{\left|\mathrm{b}\right|}^2 \end{gathered}$$

$$\begin{gathered} {\left|a\right|}^2+\frac{{\cos }^2{\displaystyle \frac{胃}{2}}}{{\sin }^2{\displaystyle \frac{胃}{2}}}{\left|a\right|}^2=1 \\ {\left|a\right|}^2\left[1+\frac{{\cos }^2{\displaystyle \frac{胃}{2}}}{{\sin }^2{\displaystyle \frac{胃}{2}}}\right]=1 \\ a=e^{-i桅}\sin \left(\frac{胃}{2}\right) \end{gathered}$$

Substitute the above value in$b=-a{e}^{i蠒}\frac{\cos {\displaystyle \frac{胃}{2}}}{\sin {\displaystyle \frac{胃}{2}}}$.

$$\begin{gathered} b=-\left(e^{-i桅}\sin \left(\frac{胃}{2}\right)\right)e^{-i蠒}\frac{\cos {\displaystyle \frac{胃}{2}}}{\sin {\displaystyle \frac{胃}{2}}} \\ =-e^{i蠒}\cos \frac{胃}{2} \end{gathered}$$

Substitute the values of a, and b in $\left(\begin{array}{c}a\\ b\end{array}\right)$.

$\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}{e}^{-i桅}\sin \frac{胃}{2}\\ -\cos \frac{胃}{2}\end{array}\right)$

Thus, the eigen values and the eigen spinors are role="math" localid="1659015532483" $\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}\cos \frac{胃}{2}\\ e^{-i桅}\sin \frac{胃}{2}\end{array}\right)$and

$\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}{e}^{-i桅}\sin \frac{胃}{2}\\ -\cos \frac{胃}{2}\end{array}\right)$.