<<PhysicalConstants`
<<Units`

Convert[(SpeedOfLight/(400.0 Nano Meter)),Hertz]

cplus[j_,m_]:=(Sqrt[j(j+1)-m(m+1)]);
cminus[j_,m_]:=(Sqrt[j(j+1)-m(m-1)]);
ejx[j_,mp_,m_]:=((cplus[j,m])/2)KroneckerDelta[mp,m+1]+((cminus[j,m])/2)KroneckerDelta[mp,m-1];
ejy[j_,mp_,m_]:=((cplus[j,m])/(2 I))KroneckerDelta[mp,m+1]-((cminus[j,m])/(2I))KroneckerDelta[mp,m-1];
ejz[j_,mp_,m_]:=m KroneckerDelta[mp,m];

j=1/2;
Jx=Table[ℏ ejx[j,mp,m],{mp,-j,j},{m,-j,j}];
Jy=Table[ℏ ejy[j,mp,m],{mp,-j,j},{m,-j,j}];
Jz=Table[ℏ ejz[j,mp,m],{mp,-j,j},{m,-j,j}];
J2=Jx.Jx+Jy.Jy+Jz.Jz;

J2.Jy-Jy.J2//MatrixForm

Eigenvalues[Jx]

ev=Eigenvectors[Jx]

evn={};
Do[
evn=Append[evn,Normalize[ev[[i]]]];
Print[i," ",evn];
,{i,1,Dimensions[ev][[1]]}]

n={Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]};
J={Jx,Jy,Jz};
Expand[Jn=n.J/.{Sin[ϕ]->((Exp[I ϕ]-Exp[-I ϕ])/(2 I)),Cos[ϕ]->((Exp[I ϕ]+Exp[-I ϕ])/2)}]

Eigenvalues[Jn]

evJn=Eigenvectors[Jn]

evJnn={};
Do[
evJnn=Append[evJnn,Normalize[evJn[[i]]]];
,{i,1,Dimensions[evJn][[1]]}];

Limit[Simplify[evJnn,Assumptions->{θ∈Reals,ϕ∈Reals}],θ->0]

j=1;
Jx=Table[ℏ ejx[j,mp,m],{mp,-j,j},{m,-j,j}];
Jy=Table[ℏ ejy[j,mp,m],{mp,-j,j},{m,-j,j}];
Jz=Table[ℏ ejz[j,mp,m],{mp,-j,j},{m,-j,j}];
J2=Jx.Jx+Jy.Jy+Jz.Jz;

n={Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]}/.{ϕ->0}
J={Jx,Jy,Jz};

Jn=n.J;

Eigenvalues[Jn]

v={0,0,1};

Jz.v

R=MatrixExp[-I (Jy/ℏ)θ]

R.v/.{θ->π/2}

Eigenvalues[Jx]
Normalize[Eigenvectors[Jx][[3]]]