- #1
Lindsayyyy
- 219
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Hi all
Given is a Hermitian Operator H
[tex]H= \begin{pmatrix}
a & b \\
b & -a
\end{pmatrix}[/tex]
where as [tex] a=rcos \phi , b=rsin \phi[/tex]
I shall find the Eigen values as well as the Eigenvectors. Furthermore I shall show that the normalized quantum states are:
[tex] \mid + \rangle =\begin{pmatrix}
\frac{cos \phi}{2} \\
\frac{sin \phi}{2}
\end{pmatrix}[/tex]
and
[tex] \mid - \rangle =\begin{pmatrix}
\frac{-sin \phi}{2} \\
\frac{cos \phi}{2}
\end{pmatrix}[/tex]
I know that: [tex] \frac {tan \phi}{2} = \frac {1-cos \phi}{sin \phi}[/tex]
-
I calculated the Eigen values vie the determinant (=0) (let's call them lambda). I think I did that right and the solutions are:
[tex] \lambda_1=r ,\lambda_2=-r[/tex]
Furthermore I calculated two Eigenvectors. I have something like an inner product
from H times a vector, so I just swapped the two entries and put a minus in front of one. My two Eigenvectors are:
[tex] \vec v_1 =\begin{pmatrix}
-rsin \phi \\
rcos \phi - r
\end{pmatrix}
\vec v_2 =\begin{pmatrix}
-rsin \phi \\
rcos \phi + r
\end{pmatrix}[/tex]
I calculated the norm which is
[tex] ||v_1||^2 = 2r^2(1-cos \phi)[/tex]
But now I'm stuck. I don't get the solution I should get. Did I do something wrong?
Thanks for your help.
Homework Statement
Given is a Hermitian Operator H
[tex]H= \begin{pmatrix}
a & b \\
b & -a
\end{pmatrix}[/tex]
where as [tex] a=rcos \phi , b=rsin \phi[/tex]
I shall find the Eigen values as well as the Eigenvectors. Furthermore I shall show that the normalized quantum states are:
[tex] \mid + \rangle =\begin{pmatrix}
\frac{cos \phi}{2} \\
\frac{sin \phi}{2}
\end{pmatrix}[/tex]
and
[tex] \mid - \rangle =\begin{pmatrix}
\frac{-sin \phi}{2} \\
\frac{cos \phi}{2}
\end{pmatrix}[/tex]
I know that: [tex] \frac {tan \phi}{2} = \frac {1-cos \phi}{sin \phi}[/tex]
Homework Equations
-
The Attempt at a Solution
I calculated the Eigen values vie the determinant (=0) (let's call them lambda). I think I did that right and the solutions are:
[tex] \lambda_1=r ,\lambda_2=-r[/tex]
Furthermore I calculated two Eigenvectors. I have something like an inner product
from H times a vector, so I just swapped the two entries and put a minus in front of one. My two Eigenvectors are:
[tex] \vec v_1 =\begin{pmatrix}
-rsin \phi \\
rcos \phi - r
\end{pmatrix}
\vec v_2 =\begin{pmatrix}
-rsin \phi \\
rcos \phi + r
\end{pmatrix}[/tex]
I calculated the norm which is
[tex] ||v_1||^2 = 2r^2(1-cos \phi)[/tex]
But now I'm stuck. I don't get the solution I should get. Did I do something wrong?
Thanks for your help.