A single particle quantum state [tex]\Psi[/tex] is given a by a wave function (vector) representable as a linear superposition of eigenvectors [tex]\Upsilon[/tex] weighted by a number proportional to their probability of occurrence: [tex]\Psi[/tex]=a_1 [tex]\Upsilon_1[/tex]+a_2 [tex]\Upsilon_2[/tex]+..... In the case of the observable "spin", there are only two eigenvectors, which constitute a 2 dimensional basis. In the case of the observable energy or momentum, there are more than 2 eigenvectors (n-dimensional basis) What determines which basis is used in representing the state [tex]\Psi[/tex]? Is it just a matter or choice, based on which observable we want to know? Also, if [tex]\Psi[/tex] is represented in with two different basis sets, then the components of the sets must be derivable from each other....but it seems hard to be able to express all the energy, momentum, position eigenvector in terms of only the two spin eigenvectors... thanks!