- #1
LagrangeEuler
- 717
- 20
Homework Statement
Form unitary matrix from eigen vectors of ##\sigma_y## and using that unitary matrix diagonalize ##\sigma_y##.
[tex]
\sigma_y=
\begin{bmatrix}
0 & -i & \\
i & 0 & \\
\end{bmatrix}[/tex][/B]
Homework Equations
Eigen vectors of ##\sigma_y## are
[tex]
\vec{X}_1=\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 \\
i \\
\end{bmatrix}[/tex]
and
[tex]
\vec{X}_2=\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 \\
-i \\
\end{bmatrix}[/tex]
The Attempt at a Solution
I am not sure, where I am making a mistake. Unitary matrix ##U## is defined by
[tex]
U=\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 & \\
i & -i & \\
\end{bmatrix}[/tex]
whereas
[tex]
U^{\dagger}=\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & -i & \\
1 & i & \\
\end{bmatrix}[/tex].
And somehow
##U \sigma_y U^{\dagger}##
is not diagonal. Is there any explanation? Where I am making a mistake?
[/B]