A Modern Approach to Quantum Mechanics · Problem 2.4

The key is recognizing that eigenstates of different spin components are superpositions of each other—you're essentially rotating your basis. Since $|{+x}\rangle$ and $|{-x}\rangle$ are eigenstates of the $\sigma_x$ operator, you can express them as linear combinations of the $|{+y}\rangle$ and $|{-y}\rangle$ eigenstates. Use the rotation relationship between spin bases or apply the $\sigma_x$ eigenvalue equation in the $y$-basis to find the coefficients, then write each state as a column vector with components in the $y$-basis.

The key is recognizing that eigenstates of different spin components are superpositions of each other—you're essentially rotating your basis. Since $|{+x}\rangle$ and $|{-x}\rangle$ are eigenstates of the $\sigma_x$ operator, you can express them as linear combinations of the $|{+y}\rangle$ and $|{-y}\rangle$ eigenstates. Use the rotation relationship between spin bases or apply the $\sigma_x$ eigenvalue equation in the $y$-basis to find the coefficients, then write each state as a column vector with components in the $y$-basis.