"Spin operators from scratch" xp = (1,1) / sqrt(2) xm = (1,-1) / sqrt(2) yp = (1,i) / sqrt(2) ym = (1,-i) / sqrt(2) zp = (1,0) zm = (0,1) Sx = 1/2 hbar ((0,1),(1,0)) Sy = 1/2 hbar ((0,-i),(i,0)) Sz = 1/2 hbar ((1,0),(0,-1)) check(Sx == 1/2 hbar (outer(xp,conj(xp)) - outer(xm,conj(xm)))) check(Sy == 1/2 hbar (outer(yp,conj(yp)) - outer(ym,conj(ym)))) check(Sz == 1/2 hbar (outer(zp,conj(zp)) - outer(zm,conj(zm)))) "ok" Run