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Homework Statement
Find the normalised eigenspinors and eigenvalues of the spin operator Sy for a spin 1⁄2 particle
If X+ and X- represent the normalised eigenspinors of the operator Sy, show that X+ and X- are orthogonal.
Homework Equations
det | Sy - λI | = 0
Sy = ## ħ/2 \begin{bmatrix}
0 & -i \\
i & 0 \\
\end{bmatrix} ##
The Attempt at a Solution
## det | ħ/2 \begin{bmatrix}
0 & -i \\
i & 0 \\
\end{bmatrix} ## - ## \begin{bmatrix}
λ & 0 \\
0 & λ \\
\end{bmatrix} | = 0 ##
skipping a few steps here but the eigenvalues = ±ħ/2
normalised eigenspinors
## ħ/2 \begin{bmatrix}
0 & -i \\
i & 0 \\
\end{bmatrix} \begin{bmatrix}
a \\
b \\
\end{bmatrix} = ± ħ/2
\begin{bmatrix}
a \\
b \\
\end{bmatrix} ##
## ħ/2 \begin{bmatrix}
0 & -i \\
i & 0 \\
\end{bmatrix} \begin{bmatrix}
1 \\
γ \\
\end{bmatrix} = ħ/2
\begin{bmatrix}
1 \\
γ \\
\end{bmatrix} ##
## \begin{bmatrix}
-iγ \\
i \\
\end{bmatrix} = \begin{bmatrix}
1 \\
γ \\
\end{bmatrix} ##
γ = i ⇒ X+ = 1/√2 ## \begin{bmatrix}
1 \\
i \\
\end{bmatrix} ##
## ħ/2 \begin{bmatrix}
0 & -i \\
i & 0 \\
\end{bmatrix} \begin{bmatrix}
1 \\
γ \\
\end{bmatrix} = -ħ/2
\begin{bmatrix}
1 \\
γ \\
\end{bmatrix} ##
## \begin{bmatrix}
-iγ \\
i \\
\end{bmatrix} =
\begin{bmatrix}
-1 \\
-γ \\
\end{bmatrix} ##
γ = -i ⇒ X- = 1/√2 ## \begin{bmatrix}
1 \\
-i \\
\end{bmatrix} ##
Eigenvalues of the spin operator Sy = ±ħ/2
Normalised eigenspinors =
X+ = 1/√2 ## \begin{bmatrix}
1 \\
i \\
\end{bmatrix} ##
and
X- = 1/√2 ## \begin{bmatrix}
1 \\
-i \\
\end{bmatrix} ##I've got the eigenvalues and normalised eigenspinors but I'm not sure how to show the eigenspinors are orthogonal.