I was reading a web page (http://electron6.phys.utk.edu/qm1/modules/m12/spinor.htm) that claims that the state vector of a spin-1/2 particle is completely specified by a two-component spinor, just as the state vector of a spinless particle is completely specified by its components in position space. This confused me greatly - it seems to me that the complete description of a spin-1/2 particle would be a state vector with components in the tensor product of both the "position space" and the "spin space" (the latter being two-dimensional). How do you get a "complete description" by collapsing all this information into a spinor with two complex components, and how do you extract the spatial coordinates of the particle from the spinor? By "spatial coordinates," I mean the components of the particle's state vector in that tensor product space I mentioned. I'm trying to understand this as a step toward understanding the Dirac and Pauli equations. It seems very straightforward that the Schrodinger equation generates a time evolution of the position-dependent wavefunction for a spinless particle, but it's not at all clear to me how these former two equations generate such a time evolution for a particle with spin when they seem to concern only the spin components of the wavefunction, and not the spatial components.