Introduction to Quantum Mechanics · Problem A.18

Afterword — David J. Griffiths

Physics — Quantum Mechanics · eigenvectors-eigenvalues

Problem

The 2×22 \times 2 matrix representing a rotation of the xyxy plane is T=(cos⁡θ−sin⁡θsin⁡θcos⁡θ).\mathsf{T} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}. Show that (except for certain special angles---what are they?) this matrix has no real eigenvalues. (This reflects the geometrical fact that no vector in the plane is carried into itself under such a rotation; contrast rotations in three dimensions.) This matrix does, however, have complex eigenvalues and eigenvectors. Find them. Construct a matrix S\mathsf{S} that diagonalizes T\mathsf{T}. Perform the similarity transformation (STS−1)\left(\mathsf{S}\mathsf{T}\mathsf{S}^{-1}\right) explicitly, and show that it reduces T\mathsf{T} to diagonal form.

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