Angular momentum  and rotation operator in quantum mechanics 
(Mathematica Program)
Eigenvalues and eigenkets
Matatsugu Sei Suzuki and Itsuko S. Suzuki
Department of Physics, SUNY at Binghamton, 
Binghamton, New York, U.S.A.

Angular momentum {1/2, 1, 3/2, 2, 5/2, 3}
Eigen values and eigenkets
Unitary operator U
Transpose of U: UT
Its Hermitian conjugate of U:   UH=(SuperPlus[U])
(SuperStar[U]);	Com[plex conjugate
Rotation operator Rz(ϕ)Ry(θ)
Commutation relations such that [Jx,Jy]=iℏ Jz
((Note) We use ℏ = 1 (units) for simplicity
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(1)	Matrix 		j = 1/2

Clear["Global`*"];j=1/2;(SuperStar[exp_ ]):=exp/.{Complex[re_,im_]:>Complex[re,-im]};Jx[j_,n_,m_]:=(1/2)(Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2)(Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jy[j_,n_,m_]:=-(1/2)I  (Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2) I (Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jz[j_,n_,m_]:=1 m KroneckerDelta[n,m];Jx=Table[Jx[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jy=Table[Jy[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jz=Table[Jz[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];

Jx//MatrixForm

Jy//MatrixForm

Jz//MatrixForm

Eigenkets and eigenvalues of Jx (J = 1/2)

eq1=Eigensystem[Jx]

ψ1x=Normalize[eq1[[2,2]]];ψ1x//MatrixForm

ψ2x=-Normalize[eq1[[2,1]]];ψ2x//MatrixForm

UxT={ψ1x,ψ2x};Ux=Transpose[UxT]; Ux//MatrixForm

UxH=(SuperStar[UxT]);UxH//MatrixForm

UxH.Ux

Eigenkets and eigenvalues of Jy (J = 1/2)

eq2=Eigensystem[Jy]

ψ1y=I Normalize[eq2[[2,2]]]; ψ1y//MatrixForm

ψ2y=-I  Normalize[eq2[[2,1]]];ψ2y//MatrixForm

UyT={ψ1y,ψ2y};Uy=Transpose[UyT]; Uy//MatrixForm

UyH=(SuperStar[UyT]);UyH//MatrixForm

UyH.Uy//Simplify

Uy.UyH//Simplify

Jx.Jy-Jy.Jx-I  Jz

Jy.Jz-Jz.Jy-I  Jx

Jx.Jx+Jy.Jy+Jz.Jz//MatrixForm

Jx.Jx-Jy.Jy//Simplify//MatrixForm

Jx+I Jy//MatrixForm

Jx-I Jy//MatrixForm

Rotation operator (J=1/2)

R=MatrixExp[-I Jz ϕ].MatrixExp[-I Jy θ]//FullSimplify; R//MatrixForm

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(2)	Matrix 		j = 1

Clear["Global`*"];j=1;(SuperStar[exp_ ]):=exp/.{Complex[re_,im_]:>Complex[re,-im]};Jx[j_,n_,m_]:=(1/2)(Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2)(Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jy[j_,n_,m_]:=-(1/2)I  (Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2) I (Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jz[j_,n_,m_]:=1 m KroneckerDelta[n,m];Jx=Table[Jx[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jy=Table[Jy[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jz=Table[Jz[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];

Jx//MatrixForm

Jy//MatrixForm

Jz//MatrixForm

Eigenkets and eigenvalues of Jx (J= 1)

eq1=Eigensystem[Jx]

ψ1x=Normalize[eq1[[2,2]]];ψ1x//MatrixForm

ψ2x=-Normalize[eq1[[2,3]]];ψ2x//MatrixForm

ψ3x=Normalize[eq1[[2,1]]];ψ3x//MatrixForm

UxT={ψ1x,ψ2x,ψ3x};Ux=Transpose[UxT]; Ux//MatrixForm

UxH=(SuperStar[UxT]);UxH//MatrixForm

UxH.Ux

Ux.UxH

Eigenkets and eigenvalues of Jy (J = 1)

eq2=Eigensystem[Jy]

ψ1y=-Normalize[eq2[[2,2]]]; ψ1y//MatrixForm

ψ2y=Normalize[eq2[[2,3]]];ψ2y//MatrixForm

ψ3y=- Normalize[eq2[[2,1]]];ψ3y//MatrixForm

UyT={ψ1y,ψ2y,ψ3y};Uy=Transpose[UyT]; Uy//MatrixForm

UyH=(SuperStar[UyT]);UyH//MatrixForm

UyH.Uy//Simplify

Uy.UyH//Simplify

Jx.Jy-Jy.Jx-I  Jz

Jy.Jz-Jz.Jy-I   Jx

Jx+I Jy//MatrixForm

Jx-I Jy//MatrixForm

Jx.Jx-Jy.Jy//Simplify//MatrixForm

Rotation operator (J=1)

R=MatrixExp[-I Jz ϕ].MatrixExp[-I Jy θ]//FullSimplify; R//MatrixForm

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(3) 	Matrix 		j = 3/2

Clear["Global`*"];j=3/2;(SuperStar[exp_ ]):=exp/.{Complex[re_,im_]:>Complex[re,-im]};Jx[j_,n_,m_]:=(1/2)(Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2)(Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jy[j_,n_,m_]:=-(1/2)I  (Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2) I (Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jz[j_,n_,m_]:=1 m KroneckerDelta[n,m];Jx=Table[Jx[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jy=Table[Jy[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jz=Table[Jz[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];

Jx//MatrixForm

Jy//MatrixForm

Jz//MatrixForm

Eigenkets and eigenvalues of Jx (J=3/2)

eq1=Eigensystem[Jx]

ψ1x=Normalize[eq1[[2,2]]];ψ1x//MatrixForm

ψ2x=-Normalize[eq1[[2,4]]];ψ2x//MatrixForm

ψ3x=Normalize[eq1[[2,3]]];ψ3x//MatrixForm

ψ4x=-Normalize[eq1[[2,1]]];ψ4x//MatrixForm

UxT={ψ1x,ψ2x,ψ3x, ψ4x};Ux=Transpose[UxT]; Ux//MatrixForm

UxH=(SuperStar[UxT]);UxH//MatrixForm

UxH.Ux

Ux.UxH

Eigenkets and eigenvalues of Jy (J=3/2)

eq2=Eigensystem[Jy]

ψ1y=-I Normalize[eq2[[2,2]]]; ψ1y//MatrixForm

ψ2y=I Normalize[eq2[[2,4]]];ψ2y//MatrixForm

ψ3y=- I Normalize[eq2[[2,3]]];ψ3y//MatrixForm

ψ4y=I Normalize[eq2[[2,1]]];ψ4y//MatrixForm

UyT={ψ1y,ψ2y,ψ3y, ψ4y};Uy=Transpose[UyT]; Uy//MatrixForm

UyH=(SuperStar[UyT]);UyH//MatrixForm

UyH.Uy//Simplify

Uy.UyH//Simplify

Jx.Jy-Jy.Jx-I  Jz

Jy.Jz-Jz.Jy-I   Jx

Jx.Jx+Jy.Jy+Jz.Jz//MatrixForm

Jx+I Jy//Simplify//MatrixForm

Jx-I Jy//Simplify//MatrixForm

Jx.Jx-Jy.Jy//Simplify//MatrixForm

Rotation operator (J=3/2)

R=MatrixExp[-I Jz ϕ].MatrixExp[-I Jy θ]//FullSimplify; R//MatrixForm

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(4)	Matrix		j = 2

Clear["Global`*"];j=2;(SuperStar[exp_ ]):=exp/.{Complex[re_,im_]:>Complex[re,-im]};Jx[j_,n_,m_]:=(1/2)(Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2)(Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jy[j_,n_,m_]:=-(1/2)I  (Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2) I (Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jz[j_,n_,m_]:=1 m KroneckerDelta[n,m];Jx=Table[Jx[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jy=Table[Jy[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jz=Table[Jz[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];

Jx//MatrixForm

Jy//MatrixForm

Jz//MatrixForm

Eigenkets and eigenvalues of Jx (J=2)

eq1=Eigensystem[Jx]

ψ1x=Normalize[eq1[[2,2]]];ψ1x//MatrixForm

ψ2x=-Normalize[eq1[[2,4]]];ψ2x//MatrixForm

ψ3x=Normalize[eq1[[2,5]]];ψ3x//MatrixForm

ψ4x=-Normalize[eq1[[2,3]]];ψ4x//MatrixForm

ψ5x=Normalize[eq1[[2,1]]];ψ5x//MatrixForm

UxT={ψ1x,ψ2x,ψ3x, ψ4x,ψ5x};Ux=Transpose[UxT]; Ux//MatrixForm

UxH=(SuperStar[UxT]);UxH//MatrixForm

UxH.Ux

Ux.UxH

Eigenkets and eigenvalues of Jy (J = 2)

eq2=Eigensystem[Jy]

ψ1y= Normalize[eq2[[2,2]]]; ψ1y//MatrixForm

ψ2y=- Normalize[eq2[[2,4]]];ψ2y//MatrixForm

ψ3y= Normalize[eq2[[2,5]]];ψ3y//MatrixForm

ψ4y= -Normalize[eq2[[2,3]]];ψ4y//MatrixForm

ψ5y= Normalize[eq2[[2,1]]];ψ5y//MatrixForm

UyT={ψ1y,ψ2y,ψ3y, ψ4y, ψ5y};Uy=Transpose[UyT]; Uy//MatrixForm

UyH=(SuperStar[UyT]);UyH//MatrixForm

UyH.Uy//Simplify

Uy.UyH//Simplify

Jx.Jy-Jy.Jx-I  Jz

Jy.Jz-Jz.Jy-I   Jx

Jx.Jx+Jy.Jy+Jz.Jz//MatrixForm

Jx+I Jy//Simplify//MatrixForm

Jx-I Jy//Simplify//MatrixForm

Jx.Jx-Jy.Jy//Simplify//MatrixForm

Rotation operator (J=2)

R=MatrixExp[-I Jz ϕ].MatrixExp[-I Jy θ]//FullSimplify; R//MatrixForm

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(5)	Matrix		j = 5/2

Clear["Global`*"];j=5/2;(SuperStar[exp_ ]):=exp/.{Complex[re_,im_]:>Complex[re,-im]};Jx[j_,n_,m_]:=(1/2)(Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2)(Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jy[j_,n_,m_]:=-(1/2)I  (Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2) I (Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jz[j_,n_,m_]:=1 m KroneckerDelta[n,m];Jx=Table[Jx[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jy=Table[Jy[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jz=Table[Jz[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];

Jx//MatrixForm

Jy//MatrixForm

Jz//MatrixForm

Jx.Jy-Jy.Jx - I Jz//Simplify//MatrixForm

Jx+I Jy//Simplify//MatrixForm

Jx-I Jy//Simplify//MatrixForm

Jx.Jx-Jy.Jy//Simplify//MatrixForm

Rotation operator (J = 5/2)

R=MatrixExp[-I Jz ϕ].MatrixExp[-I Jy θ]//FullSimplify; R//MatrixForm

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(6)	Matrix		j = 3

Clear["Global`*"];j=3;(SuperStar[exp_ ]):=exp/.{Complex[re_,im_]:>Complex[re,-im]};Jx[j_,n_,m_]:=(1/2)(Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2)(Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jy[j_,n_,m_]:=-(1/2)I  (Sqrt[(j-m) (j+m+1)]) KroneckerDelta[n,m+1]+(1/2) I (Sqrt[(j+m) (j-m+1)]) KroneckerDelta[n,m-1];Jz[j_,n_,m_]:=1 m KroneckerDelta[n,m];Jx=Table[Jx[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jy=Table[Jy[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];Jz=Table[Jz[j,n,m],{n,j,-j,-1},{m,j,-j,-1}];

Jx//MatrixForm

Jy//MatrixForm

Jz//MatrixForm

Jx.Jy-Jy.Jx - I Jz//Simplify//MatrixForm

Jx+I Jy//Simplify//MatrixForm

Jx-I Jy//Simplify//MatrixForm

Jx.Jx-Jy.Jy//Simplify//MatrixForm

Rotation operator (J = 3)

R=MatrixExp[-I Jz ϕ].MatrixExp[-I Jy θ]//FullSimplify; R//MatrixForm