Suddenly I am at a loss with something I used to think I understood! From Consistent Quantum theory, Griffiths, pg 51: Our basis is |z+>, |z-> I can write |w+> = +cos(α/2)exp(-iθ/2) |z+> + sin(α/2)exp(iθ/2) |z-> In this case, if I choose α = π/2 and θ = π, then this |w+> points in the -x direction (or so he says): |w+> = (1/√2)(-i) |z+> + (1/√2)(i) |z-> = (i/√2) (|z-> - |z+>) Can somebody remind me how this superposition of states can correspond to an orthogonal direction, x? The image in my head is of the z axis, and how no two vectors purely along the z axis can ever add up to a vector along the x axis. I am sure that I am missing the main point here of QM, Hilbert spaces, orthogonal states, and non-commuting operators. Thanks!