How can we represent a spin states
$ \lvert S_x:+\rangle, \lvert S_y:+\rangle,\lvert S_z:+\rangle ,\lvert S_x:-\rangle, \lvert S_y:-\rangle $ and $\lvert S_z:-\rangle$ in position representation like
$ \langle x\lvert S_z:+\rangle$ ?
Is there any position representation for spin operators $ S_z, S_y, S_z $?
Zettili says in his book of Quantum mechanics that "Spin cannot be described by a differential operator". Does it mean spin cannot have a position representation? He does not talk about it more.
Best Answer
Unlike the angular momentum operators which inherit their position representation from the position representation of $\hat x$ and $\hat p$: $$ \hat L_z\to -i\hbar (y\partial_x -x\partial_y) $$ it is not possible to write $\hat S_z$ in terms of the classical position and momenta because spin is an "intrinsic" rather than spatial degree of freedom: dimensional analysis shows that only products like $yp_x$ have units of angular momentum so, after accounting for the commutation relations to be respected, a position representation of spin would be more or less identical to the position representation of "ordinary" angular momentum.
Moreover, quantizing angular momentum necessarily leads to integer values of $\ell$ through the use of (spherical) coordinates see:
Thus a position-based representation for spin could not accommodate half-integer values.