Introduction to Quantum Mechanics · Problem A.15

Afterword — David J. Griffiths

Physics — Quantum Mechanics · changing-bases

Problem

In the usual basis (i^,j^,k^)\left(\hat{i}, \hat{j}, \hat{k}\right), construct the matrix Tx\mathsf{T}_x representing a rotation through angle θ\theta about the xx axis, and the matrix Ty\mathsf{T}_y representing a rotation through angle θ\theta about the yy axis. Suppose now we change bases, to i^′=j^\hat{i}' = \hat{j}, j^′=−i^\hat{j}' = -\hat{i}, k^′=k^\hat{k}' = \hat{k}. Construct the matrix S\mathsf{S} that effects this change of basis, and check that STxS−1\mathsf{S}\mathsf{T}_x\mathsf{S}^{-1} and STyS−1\mathsf{S}\mathsf{T}_y\mathsf{S}^{-1} are what you would expect.

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