ψ = (1/(Sqrt[14])) {1, 2, 
            3 I};
ψzmas = {1, 0, 0};
ψzcero = {0, 1, 0};
ψzmenos = {0, 0, 1};

Abs[Conjugate[ψzmenos] . ψ]^2

Jx = (h/(Sqrt[2])) {{0, 1, 0}, {1, 0, 1}, {0, 1, 0}};
Eigenvalues[Jx]
VV = Eigenvectors[Jx]
ψxmas = 
      VV[[1]]/(Sqrt[(VV[[1]] . VV[[1]])])
ψxcero = 
      VV[[2]]/(Sqrt[(VV[[2]] . VV[[2]])])
ψxmenos = 
      VV[[3]]/(Sqrt[(VV[[3]] . VV[[3]])])
p1 = 
      FullSimplify[
        Abs[Conjugate[ψxmas] . ψ]^2]
N[%]
p2 = 
      FullSimplify[
        Abs[Conjugate[ψxcero] . ψ]^2]
N[%]
p3 = 
      FullSimplify[
        Abs[Conjugate[ψxmenos] . ψ]^2]
N[%]
p1 + p2 + p3
N[%]