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Homework Statement
This question is from Principles of Quantum Mechanics by R. Shankar.
Given the operator (matrix) \Omega with eigenvalues e^{i\theta} and e^{-i\theta}, I am told to find the corresponding eigenvectors.
Homework Equations
<br /> \Omega = \left[ \begin{array}{cc}<br /> \cos{\theta} & \sin{\theta} \\<br /> -\sin{\theta} & \cos{\theta} \\<br /> \end{array} \right]<br />
<br /> \Omega \left[ \begin{array}{c}<br /> x_1 \\<br /> x_2 \\<br /> \end{array} \right] = \left[ \begin{array}{c}<br /> x_1 \cos{\theta} + x_2 \sin{\theta} \\<br /> -x_1 \sin{\theta} + x_2 \cos{\theta} \\<br /> \end{array}<br /> \right] <br />
<br /> e^{i\theta} \left[ \begin{array}{c}<br /> x_1 \\<br /> x_2 \\<br /> \end{array} \right] = \left[ \begin{array}{c}<br /> x_1 \cos{\theta} + x_1 i\sin{\theta} \\<br /> x_2 \cos{\theta} + x_2 i\sin{\theta} \\<br /> \end{array}\right]<br />
The Attempt at a Solution
I let the matrix operate on the generic vector (x_1, x_2)^T and demand that the resulting vector is equal to (e^{i\theta}x_1, e^{i\theta}x_2)^T. From this i get the condition that x_2 = ix_2and x_1 = -ix_2, which implies that x_1 = x_2 = 0. Am i missing something crucial?
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