Smith has shown above that any state

where

is an eigenstate of some operator

where

. But since this also holds true for all complex multiples of the operator or the eigenstate, we can conclude that

every state

is an eigenstate of the angular momentum operator

along the (not necessarily unit) vector

in real 3-dimensional space. (Recall

, etc).

That completes part (i) of the problem.

For part (ii) we note that since

is a representation space of SO(3), we can

rotate the spin-up state

to point in any direction (by applying the appropriate rotation

from SO(3)). If

represents the set of physically distinguishable eigenstates (i.e. we consider complex multiples of

to represent the same thing), then there must be an element of

corresponding to each possible rotated spin state

. But

is isomorphic to the Riemann sphere, i.e. to exactly the set of all possible directions in 3-space; so a topology preserving mapping

,

SO(3) must therefore be

onto ; and hence

every element of

(and hence every spin state

) corresponds to some rotation

of the spin-up state... i.e. to a spin eigenstate along

some axis in 3-space.