Spinors 
Calculations of the Appendix 3.3 of the Lecture Notes of the Course
			Kinematical Theory of Elementary Spinning  Particles


 Spinors of spin 1/2 in the representation as eigenvectors of  W3 and T3  according to the  Notes (3.70)-(3.73) 
**********************************************)

Φ1=(Sqrt[2])(Cos[α/2]+I Cos[θ] Sin[α/2]);
Φ2=I(Sqrt[2])Sin[α/2] Sin[θ] E^(-I ϕ);
Φ3=I (Sqrt[2])Sin[α/2] Sin[θ] E^(I ϕ);
Φ4=(Sqrt[2])(Cos[α/2]-I Cos[θ] Sin[α/2]);

(* Their complex conjugates *)

Φ1c=(Sqrt[2])(Cos[α/2]-I Cos[θ] Sin[α/2]);
Φ2c=-I(Sqrt[2])Sin[α/2] Sin[θ] E^(I ϕ);
Φ3c=-I (Sqrt[2])Sin[α/2] Sin[θ] E^(-I ϕ);
Φ4c=(Sqrt[2])(Cos[α/2]+I Cos[θ] Sin[α/2]);

(******************************************************
 Spin operators on the orientation variables W1, W2, W3  (p.136)
and (4.74) - (4.79)
*******************************************************)

w1[f_]:=(2 Sin[θ] Cos[ϕ] D[f,α]+(Cos[θ] Cos[ϕ]/Tan[α/2]-Sin[ϕ])D[f,θ]-(Sin[ϕ]/Tan[α/2]/Sin[θ]+Cos[θ] Cos[ϕ]/Sin[θ]) D[f,ϕ])/(2I);
w2[f_]:=(2 Sin[θ] Sin[ϕ] D[f,α]+(Cos[θ] Sin[ϕ]/Tan[α/2]+Cos[ϕ])D[f,θ]-(-Cos[ϕ]/Tan[α/2]/Sin[θ]+Cos[θ] Sin[ϕ]/Sin[θ]) D[f,ϕ])/(2I);
w3[f_]:=(2 Cos[θ]  D[f,α]-Sin[θ]/Tan[α/2]D[f,θ]+ D[f,ϕ])/(2I);

Action of W3 on the above basis  as eigenvectors of spin 1/2 y -1/2
w3[{Φ1,Φ2,Φ3,Φ4}]-{Φ1,-Φ2,Φ3,-Φ4}/2//TrigExpand

(******************************************************
Spin operators on the orientation squared, W^2 =T^2,  and action on the above basis as eigenvectores of value 1/2  (p.136)
*******************************************************)

wQ[f_]:=-(D[f,{α,2}]+1/Tan[α/2]D[f,α]+1/Sin[α/2]^2/4(D[f,{θ,2}]+Cos[θ]/Sin[θ] D[f,θ]+1/Sin[θ]^2D[f,{ϕ,2}]));
wQ[{Φ1,Φ2,Φ3,Φ4}]-{Φ1,Φ2,Φ3,Φ4}(1/2+1)/2//TrigFactor

(*******************************************************
Spin projection operators on the body axis T1,T2,T3, (3.53) . 
in the normal parameterization and in (4.80) - (4.82) 
************************************************)

t1[f_]:=(2 Sin[θ] Cos[ϕ] D[f,α]+(Cos[θ] Cos[ϕ]/Tan[α/2]+Sin[ϕ])D[f,θ]-(Sin[ϕ]/Tan[α/2]/Sin[θ]-Cos[θ] Cos[ϕ]/Sin[θ]) D[f,ϕ])/(2I);
t2[f_]:=(2 Sin[θ] Sin[ϕ] D[f,α]+(Cos[θ] Sin[ϕ]/Tan[α/2]-Cos[ϕ])D[f,θ]-(-Cos[ϕ]/Tan[α/2]/Sin[θ]-Cos[θ] Sin[ϕ]/Sin[θ]) D[f,ϕ])/(2I);
t3[f_]:=(2 Cos[θ]  D[f,α]-Sin[θ]/Tan[α/2]D[f,θ]- D[f,ϕ])/(2I);

Action of T3 on the above basis as eigenvectors of spin 1/2 and -1/2
t3[{Φ1,Φ2,Φ3,Φ4}]-{Φ1,Φ2,-Φ3,-Φ4}/2//TrigExpand

(*******************************************************
Normalized measure (3.66) and orthogonality of the function 1 and the Φi
***********************************************)
Integrate[Sin[α/2]^2 Sin[θ]/(4 π^2) ,{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]
Integrate[ {Φ1,Φ2,Φ3,Φ4}Sin[α/2]^2 Sin[θ]/(4 π^2) ,{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

(*******************************************************
Orthogonality and normalization of the spinors Φi, in the normalized measure (3.66)
***********************************************)
Integrate[Φ1c {Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

Integrate[Φ2c {Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

Integrate[Φ3c {Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

Integrate[Φ4c {Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

(*******************************************************
Matrix representation of the spin operators Si=Wi in the basis Φi   (3.74)
*************************************************)

{Integrate[Φ1c  w1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c  w1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  w1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  w1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  w2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c  w2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  w2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  w2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  w3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c  w3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  w3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  w3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

(*******************************************************
Matrix representation of the spin operators Ti in the basis Φi   (3.75)
*******************************************************)

{Integrate[Φ1c  t1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c  t1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  t1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  t1[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  t2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c  t2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  t2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  t2[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  t3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c  t3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  t3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  t3[{Φ1,Φ2,Φ3,Φ4}] Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

(*******************************************************
 Components of the unit vectors of the body frame  (p.131)  
*******************************************************)

e11=Cos[α]+Sin[θ]^2 Cos[ϕ]^2(1-Cos[α]);
e12=Cos[θ]Sin[α]+Sin[θ]^2 Sin[ϕ] Cos[ϕ](1-Cos[α]);
e13=-Sin[θ]Sin[ϕ]Sin[α]+Sin[θ] Cos[θ] Cos[ϕ](1-Cos[α]);
e21=-Cos[θ]Sin[α]+Sin[θ]^2 Sin[ϕ] Cos[ϕ](1-Cos[α]);
e22=Cos[α]+Sin[θ]^2 Sin[ϕ]^2(1-Cos[α]);
e23=Sin[θ]Cos[ϕ]Sin[α]+Sin[θ] Cos[θ] Sin[ϕ](1-Cos[α]);
e31=Sin[θ]Sin[ϕ]Sin[α]+Sin[θ] Cos[θ] Cos[ϕ](1-Cos[α]);
e32=-Sin[θ]Cos[ϕ]Sin[α]+Sin[θ] Cos[θ] Sin[ϕ](1-Cos[α]);
e33=Cos[α]+Cos[θ]^2(1-Cos[α]);

(*******************************************************
Matrix representation of the components of the unit vectors of the body frame (3.76) in the basis Φi.
****************************************************)

{Integrate[Φ1c  e11{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e11{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e11{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e11{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e12{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e12{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e12{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e12{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e13{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e13{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e13{Φ1,Φ2,Φ3,Φ4}Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e13{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e21{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e21{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e21{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e21{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e22{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e22{Φ1,Φ2,Φ3,Φ4}Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e22{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e22{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e23{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e23{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e23{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e23{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e31{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e31{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e31{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e31{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e32{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e32{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e32{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e32{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Φ1c  e33{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ2c e33{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ3c  e33{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Φ4c  e33{Φ1,Φ2,Φ3,Φ4} Sin[α/2]^2 Sin[θ] /(4 π^2),{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

(****************************************** 
Action of T- =T1-iT2 and T+=T1+iT2 on the spinors of the basis Φi  (p.138)
****************************************************)
{t1[Φ1]-I t2[Φ1],
t1[Φ2]-I t2[Φ2],
t1[Φ3]-I t2[Φ3]-Φ1,
t1[Φ4]-I t2[Φ4]-Φ2}//TrigExpand
{t1[Φ3]+I t2[Φ3],
t1[Φ4]+I t2[Φ4],
t1[Φ1]+I t2[Φ1]-Φ3,
t1[Φ2]+I t2[Φ2]-Φ4}//TrigExpand

(*************************************
Action of W- =W1-iW2 and W+=W1+iW2 on the spinors of the basis Φi  (p.138)
*****************************************)
{w1[Φ1]-I w2[Φ1]-Φ2,
w1[Φ3]-I w2[Φ3]-Φ4,
w1[Φ2]-I w2[Φ2],
w1[Φ4]-I w2[Φ4]}//TrigExpand
{w1[Φ1]+I w2[Φ1],
w1[Φ3]+I w2[Φ3],
w1[Φ2]+I w2[Φ2]-Φ1,
w1[Φ4]+I w2[Φ4]-Φ3}//TrigExpand

(***************************************
Spherical Harmonics on the orientation of the velocity Ylm(β,λ) (3.79-3.81)
*********************************)

Y11=-(Sqrt[3/2])Sin[β]E^(I λ);
Y10=(Sqrt[3])Cos[β];
Y1m=(Sqrt[3/2])Sin[β]E^(-I λ);

(* Their complex conjugates *)

Y11c=-(Sqrt[3/2])Sin[β]E^(-I λ);
Y10c=(Sqrt[3])Cos[β];
Y1mc=(Sqrt[3/2])Sin[β]E^(I λ);

(*******************************************************
 Normalized invariant measure of the Spherical Harmonics (3.58)
*******************************************************)
Integrate[ Sin[β]/(4 Pi),{β,0,Pi},{λ,0,2 Pi}]

(*******************************************************
 Orthogonality and normalization of the Spherical Harmonics  
*******************************************************)

Integrate[Y11c {Y11,Y10,Y1m} Sin[β]/(4 Pi),{β,0,Pi},{λ,0,2 Pi}]

Integrate[Y10c {Y11,Y10,Y1m} Sin[β]/(4 Pi),{β,0,Pi},{λ,0,2 Pi}]

Integrate[Y1mc {Y11,Y10,Y1m} Sin[β]/(4 Pi),{β,0,Pi},{λ,0,2 Pi}]

(*******************************************************
 Spin operators on the orientation of the velocity (β,λ), Z1,Z2,Z3 , (3.56)
*******************************************************)

z1[f_]:=(Sin[λ] D[f,β]+Cos[β]/ Sin[β] Cos[λ]D[f,λ])I;
z2[f_]:=-(Cos[λ] D[f,β]-Cos[β]/ Sin[β] Sin[λ]D[f,λ])I;
z3[f_]:=-D[f,λ]I;
zQ[f_]:=-(D[f,{β,2}]+Cos[β]/Sin[β]D[f,β]+1/Sin[β]^2 D[f,{λ,2}]);

(*******************************************************
Spin operadores with respect to the center of charge S1=Z1+W1, etc
******************************************************)

s1[f_]:=z1[f]+w1[f];
s2[f_]:=z2[f]+w2[f];
s3[f_]:=z3[f]+w3[f];

(*******************************************************
Spin 1/2 spinors as eigenvectors of S^2 =1/2, S3 and T3.
They are the vectors Ψi (3.82)-(3.85)  and  (4.87)-(4.90)
*******************************************************)
Ψ1=(Y10 Φ1 -(Sqrt[2]) Y11 Φ2)/(Sqrt[3]);
Ψ2=(-Y10 Φ2 +(Sqrt[2]) Y1m Φ1)/(Sqrt[3]);
Ψ3=(Y10 Φ3 -(Sqrt[2]) Y11 Φ4)/(Sqrt[3]);
Ψ4=(-Y10 Φ4 +(Sqrt[2]) Y1m Φ3)/(Sqrt[3]);

(* Their complex conjugates *)

Ψ1c=(Y10c Φ1c -(Sqrt[2]) Y11c Φ2c)/(Sqrt[3]);
Ψ2c=(-Y10c Φ2c +(Sqrt[2]) Y1mc Φ1c)/(Sqrt[3]);
Ψ3c=(Y10c Φ3c -(Sqrt[2]) Y11c Φ4c)/(Sqrt[3]);
Ψ4c=(-Y10c Φ4c +(Sqrt[2]) Y1mc Φ3c)/(Sqrt[3]);

(********************************************************
Invariant and normalized measure of the spinors  Ψi  (p. 172)
*******************************************************)
Integrate[ Sin[α/2]^2 Sin[θ]  Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

(********************************************************
Orthogonality and normalization of the spinors Ψi (3.82)-(3.85)
*******************************************************)
Integrate[Ψ1c {Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]  Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

Integrate[Ψ2c {Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]  Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

Integrate[Ψ3c {Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]  Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

Integrate[Ψ4c {Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]  Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

(*******************************************************
 Spin operador Z^2, and its matrix representation in the basis Ψi  (3.84) 
*******************************************************)
zQ[{Ψ1,Ψ2,Ψ3,Ψ4}]-2{Ψ1,Ψ2,Ψ3,Ψ4}//TrigFactor
{Integrate[Ψ1c  zQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  zQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  zQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  zQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

(*******************************************************
 Spin operador W^2, and its matrix representationin the basis Ψi  (3.84) 
*******************************************************)

wQ[{Ψ1,Ψ2,Ψ3,Ψ4}]-3/4{Ψ1,Ψ2,Ψ3,Ψ4}//TrigFactor
{Integrate[Ψ1c  wQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  wQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  wQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  wQ[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

Operator W3 (Ψ1 is not eigenvector of W3)
w3[ Ψ1]-Ψ1/2//TrigFactor
Integrate[Ψ1c w3[Ψ1] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]
Integrate[Ψ2c w3[Ψ1] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]

(*******************************************************
 Spin operators with respect to the center of charge S1=Z1+W1, etc, and their matrix representation in the basis Ψi , (3.86) 
*******************************************************)

{Integrate[Ψ1c  s1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  s1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  s1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  s1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  s2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]  /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  s2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β]  /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  s2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  s2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β]  /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  s3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  s3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  s3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  s3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

(*******************************************************
Spin operators of orbital nature Z1,Z2,Z3, etc  and their matrix representation in the basis Ψi,  (3.87) 
*******************************************************)

{Integrate[Ψ1c  z1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  z1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  z1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  z1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  z2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  z2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  z2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  z2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  z3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  z3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  z3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  z3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

(*******************************************************
Spin operators related to the orientation  W1,W2,W3,   and their matrix representation in the basis Ψi,  (3.87)  
******************************************************)

{Integrate[Ψ1c  w1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  w1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  w1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  w1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  w2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  w2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  w2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  w2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  w3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  w3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  w3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  w3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

******************************************************
 Spin projection operators on the body axis T1,T2,T3,  and their matrix representation in the basis Ψi, (3.88)
*****************************************************)

{Integrate[Ψ1c  t1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  t1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  t1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  t1[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  t2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  t2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  t2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  t2[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  t3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  t3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  t3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  t3[{Ψ1,Ψ2,Ψ3,Ψ4}] Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

(*******************************************************
Matrix representation of the components of the body axis in the basis Ψi,    (3.89)   
*******************************************************)
{Integrate[Ψ1c  e11{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e11{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e11{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e11{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e12{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e12{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e12{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e12{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e13{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e13{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e13{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e13{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e21{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e21{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e21{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e21{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e22{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e22{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e22{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e22{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e23{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e23{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e23{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e23{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e31{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e31{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e31{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e31{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e32{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e32{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e32{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e32{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm

{Integrate[Ψ1c  e33{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ2c  e33{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ3c  e33{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ] Sin[β]/(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}],
Integrate[Ψ4c  e33{Ψ1,Ψ2,Ψ3,Ψ4} Sin[α/2]^2 Sin[θ]Sin[β] /(16 π^3),{β,0,π},{λ, 0,2π},{ϕ,0, 2π},{θ, 0,π},{α,0,2π}]}//MatrixForm