My question is conceptual but specific. I'm self-studying Townsend's text 'A Modern Approach to Quantum Mechanics.' In Sec. 2.2 pg 33 (in case you have the book handy), he introduces rotation operators, in the context of spin states for spin-1/2 particles. He states that the rotation operator R(π/2j) (where j denotes the unit vector in the +y direction of an xyz coordinate system) rotates the [+z> ket (sorry about the clumsy notation) into the [+x> ket. He even illustrates this with a straightforward diagram of an xyz coordinate system. In the text he states the following: 'The interaction of the magnetic moment of a spin-1/2 particle with the magnetic field causes the quantum spin state of the particle to rotate about the direction of the field as time progresses. In particular, if the magnetic field points in the y direction and the particle starts out in the state [+z>, the spin will rotate in the xz plane. At some later time the particle will be in the state [+x>.' But here's what's driving me crazy: Isn't it true that these kets that are being rotated don't even live in the same space as the magnetic field? I.e. they are not spacetime vectors. In fact, Townsend himself, earlier in the text, made a point of emphasizing this point, that these kets don't represent states in ordinary 3D space but spin states in a different, 2D space altogether. So how/why is he now talking about these kets rotating around in ordinary space, relative to a magnetic field defined in ordinary space? A space where they don't even live?