- #1

timon

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## Homework Statement

This question is from

*Principles of Quantum Mechanics*by R. Shankar.

Given the operator (matrix) [itex] \Omega [/itex] with eigenvalues [itex] e^{i\theta}[/itex] and [itex] e^{-i\theta} [/itex], I am told to find the corresponding eigenvectors.

## Homework Equations

[itex]

\Omega = \left[ \begin{array}{cc}

\cos{\theta} & \sin{\theta} \\

-\sin{\theta} & \cos{\theta} \\

\end{array} \right]

[/itex]

[itex]

\Omega \left[ \begin{array}{c}

x_1 \\

x_2 \\

\end{array} \right] = \left[ \begin{array}{c}

x_1 \cos{\theta} + x_2 \sin{\theta} \\

-x_1 \sin{\theta} + x_2 \cos{\theta} \\

\end{array}

\right]

[/itex]

[itex]

e^{i\theta} \left[ \begin{array}{c}

x_1 \\

x_2 \\

\end{array} \right] = \left[ \begin{array}{c}

x_1 \cos{\theta} + x_1 i\sin{\theta} \\

x_2 \cos{\theta} + x_2 i\sin{\theta} \\

\end{array}\right]

[/itex]

## The Attempt at a Solution

I let the matrix operate on the generic vector [itex] (x_1, x_2)^T [/itex] and demand that the resulting vector is equal to [itex] (e^{i\theta}x_1, e^{i\theta}x_2)^T [/itex]. From this i get the condition that [itex] x_2 = ix_2 [/itex]and [itex] x_1 = -ix_2 [/itex], which implies that [itex] x_1 = x_2 = 0 [/itex]. Am i missing something crucial?

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