Up to this point, we have essentially presumed that the state of a system, such as an electron or even a hydrogen atom, can be specified by considering a function in space and time, a function we have called the wavefunction. That description in terms of one quantity, the scalar amplitude of the wavefunction, turns out not to be sufficient to describe quantum mechanical particles. We find that we also need to specify amplitudes for the spin of the particle.
The idea that we would need additional degrees of freedom to describe a system completely is not itself unusual. For example, in classical mechanics, we might use position as a function of time to describe an object such as ...