Introduction to Quantum Mechanics · Problem A.28

Afterword — David J. Griffiths

Physics — Quantum Mechanics · hermitian-transformations

Problem

Let T=(11−i1+i0).\mathsf{T} = \begin{pmatrix} 1 & 1-i \\ 1+i & 0 \end{pmatrix}. (a) Verify that T\mathsf{T} is hermitian. (b) Find its eigenvalues (note that they are real). (c) Find and normalize the eigenvectors (note that they are orthogonal). (d) Construct the unitary diagonalizing matrix S\mathsf{S}, and check explicitly that it diagonalizes T\mathsf{T}. (e) Check that det⁡(T)\det(\mathsf{T}) and Tr(T)\text{Tr}(\mathsf{T}) are the same for T\mathsf{T} as they are for its diagonalized form.

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