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.
