Addition of  spin angular momentum for three spin-1/2 particles, using the Kronecker products and Clebsch-Gordan coefficients

Masatsugu Sei Suzuki and Itsuko S. Suzuki
Department of Physics, SUNY at Binghamton
Binghamton, New York,  U.S.A.
(Date: March 31, 2023)

Here we discuss the eigenstates of three  spin-1/2 particles  by using two methods.

I.	Clebsch-Gordan coefficient
II.	Method of Kronecker product
III.	Comparison of our results with Schiff’s  result
IV.	Comparison of our results with Tomonaga’s  result
V.	Conclusion



The unitary operator Ua is obtained from both the Kronecker product and the Clebsch-Gordan coefficient (in this article). The unitary operator Us comes from the book of Schiff. The unitary operator Ut comes from the book of Tomonaga.
The ket |bi> is expressed as



Note that 

det(Ua)=1,	det(Us)=-1,	det(Ut)=-1

REFERENCES
1. 
L.I. Skiff, Quantum Mechanics, 3rd edition (McGraw-Hill, 1968) p.377

2. 
Sin-ItiroTomonaga , “Angular Momentum and Spin,” Supplement to Quantum Mechanics I (1962) and II  (1966) (North-Holland), edited by Susumu Kamebuchi, Yasuo Hara, and Takeyasu Kodera  (Misuzu-Shobou, 1988, in Japanese). p.46.
https://en.wikipedia.org/wiki/Shin%27ichir%C5%8D_Tomonaga

___________________________________________________________
I.	Clebsch-Gordan coefficient for three spin-1/2 particles

Addition of angular momentum with three S= 1/2 spins
D1/2 x D1/2 x D1/2 
= (D1 + D0) x D1/2 
=(D1  x D1/2 )+D1/2
= (D3/2 + D1/2) + D1/2

Clear["Global`*"];CCGG[{j1_,m1_},{j2_,m2_},{j_,m_}]:=Module[{s1},s1=If[Abs[m1]⩽j1&&Abs[m2]⩽j2&&Abs[m]⩽j,ClebschGordan[{j1,m1},{j2,m2},{j,m}],0]];

CG2[j1_,j2_,j_,a1_,a2_]:=Table[Sum[CCGG[{j1,k1},{j2,k2},{j,k1+k2}] a1[j1,k1] a2[j2,k2]  KroneckerDelta[k1+k2,m],{k1,-j1,j1},{k2,-j2,j2}],{m,-j,j}]

Addition of angular momentum with two S= 1/2 spins

j1=1/2, 	 j2=1/2		D1/2 x D1/2 = D1+D0
leading to j = 1, 0

CG2[1/2,1/2,1,b1,b2]//TableForm

CG2[1/2,1/2,0,b1,b2]//TableForm

Addition of angular momentum with three S= 1/2 spins
D1/2 x D1/2 x D1/2 
= (D1 + D0) x D1/2 
=(D1  x D1/2 )+D1/2
= (D3/2 + D1/2) + D1/2

(a)	j1=1, 	 j2=1/2	j = 3/2

j1=1;j2=1/2;j=3/2;Table[Sum[CCGG[{j1,k1},{j2,k2},{j,k1+k2}] CG2[1/2,1/2,j1,b1,b2][[k1+j1+1]]b3[j2,k2]  KroneckerDelta[k1+k2,m],{k1,-j1,j1},{k2,-j2,j2}],{m,-j,j}]//Simplify

j=3/2
|+ + +>						m=-3/2
(1/(Sqrt[3]))[|- + +> + |+ + -> + |+ - +>]		m = 1/2
(1/(Sqrt[3]))[|+ - -> + |- + -> + |- - +>]		m= - 1/2
|- - ->							m=-3/2

(b)	j1=1, 	 j2=1/2			j = 1/2

j1=1;j2=1/2;j=1/2;Table[Sum[CCGG[{j1,k1},{j2,k2},{j,k1+k2}] CG2[1/2,1/2,j1,b1,b2][[k1+j1+1]]b3[j2,k2]  KroneckerDelta[k1+k2,m],{k1,-j1,j1},{k2,-j2,j2}],{m,-j,j}]//Simplify

j=1/2
(1/(Sqrt[6]))[-|- + +> - |+ - +> + 2|+ + ->]		m = 1/2
(1/(Sqrt[6]))[|+ - -> + |- + -> - 2|- - +>]		m= - 1/2

(c)	j1=0, 	 j2=1/2			j = 1/2

j1=0;j2=1/2;j=1/2;Table[Sum[CCGG[{j1,k1},{j2,k2},{j,k1+k2}] CG2[1/2,1/2,j1,b1,b2][[k1+j1+1]]b3[j2,k2]  KroneckerDelta[k1+k2,m],{k1,-j1,j1},{k2,-j2,j2}],{m,-j,j}]//Simplify

j=1/2
(1/(Sqrt[2]))[|+ - +> - |- + +>]		m = 1/2
(1/(Sqrt[2]))[|+ - -> - |- + ->]		m= - 1/2

__________________________________________________________________________
II.	Addition of spin angular momentum for  three spin 1/2 particles  using the Kronecker product

(D1/2×D1/2)×D1/2=(D1+D0)×D1/2=D3/2+D1/2+D1/2+ (S = 3/2 and S = 1/2)

Clear["Global`*"];
(SuperStar[exp_ ]):=exp/.{Complex[re_,im_]:>Complex[re,-im]};ψ1=(({{1}, {0}}));ψ2=(({{0}, {1}}));σx=(({{0, 1}, {1, 0}}));σy=(({{0, - I}, {I, 0}}));σz=(({{1, 0}, {0, -1}}));I2=IdentityMatrix[2];

ST1=(1/2)(KroneckerProduct[σx,σx,I2]+KroneckerProduct[σy,σy, I2]
+KroneckerProduct[σz,σz,I2]+KroneckerProduct[I2,σx,σx]+KroneckerProduct[I2,σy,σy]
+KroneckerProduct[I2,σz,σz]+KroneckerProduct[σx,I2,σx]+KroneckerProduct[σy,I2, σy]
+KroneckerProduct[σz,I2,σz])+(9/4) KroneckerProduct[I2,I2,I2];

ST1//MatrixForm

eq1=Eigensystem[ST1]//Simplify

ξ1=eq1[[2,4]]; ξ2=eq1[[2,3]];ξ3=eq1[[2,2]];ξ4=eq1[[2,1]];ξ5=eq1[[2,5]];ξ6=eq1[[2,6]];ξ7=eq1[[2,7]];ξ8=eq1[[2,8]];

eq2=Orthogonalize[{ξ1,ξ2, ξ3,ξ4,ξ5,ξ6,ξ7,ξ8}]

Eigenstates

a1=eq2[[1]];
a2=eq2[[2]];
a3=eq2[[3]];
a4=eq2[[4]]; 
a5=eq2[[8]];a6=eq2[[6]];a7=-eq2[[7]];a8=-eq2[[5]];

Sz=(1/2)(KroneckerProduct[σz,I2,I2]+KroneckerProduct[I2,σz,I2]+KroneckerProduct[I2,I2,σz])

Sz//MatrixForm

D3/2 (j = 3/2, m = 3/2, 1/2,-1/2, -3/2)

ST1.a1-(15/4) a1

Sz.a1 - (3/2) a1

ST1.a2-(15/4)  a2

Sz.a2-(1/2)a2

ST1.a3-(15/4)  a3

Sz.a3+(1/2)a3

ST1.a4-(15/4)  a4

Sz.a4+(3/2)a4

D1/2 (j=1/2, m=1/2, -1/2)

ST1.a5-(3/4)  a5//Simplify

Sz.a5-(1/2)a5

ST1.a6-(3/4)  a6//Simplify

Sz.a6+(1/2)a6

D1/2 (j = 1/2, m = 1/2, -1/2)

ST1.a7-(3/4)  a7//Simplify

Sz.a7-(1/2)a7

ST1.a8-(3/4)  a8//Simplify

Sz.a8+(1/2)a8

{a1.a2,a2.a3,a3.a4,a4.a5,a5.a6,a6.a7,a7.a8}

a1//MatrixForm

a2//MatrixForm

a3//MatrixForm

a4//MatrixForm

a5//MatrixForm

a6//Simplify//MatrixForm

a7//Simplify//MatrixForm

a8//Simplify//MatrixForm

Determination of unitary operator Ua
Ua: Unitary operator
UaT: Transpose of unitary operator
UaH: Hermitain conjugate of Ua

UaT={a1, a2,a3,a4,a5,a6,a7,a8}//Simplify;

Ua=Transpose[UaT]//Simplify; UaH=(SuperStar[UaT])//Simplify; Ua//MatrixForm

UaH//MatrixForm

UaH.Ua//Simplify//MatrixForm

Det[Ua]

Basis;

b1=KroneckerProduct[ψ1,ψ1, ψ1]; b1//MatrixForm

b2=KroneckerProduct[ψ1,ψ1, ψ2];b2//MatrixForm

b3=KroneckerProduct[ψ1,ψ2, ψ1]; b3//MatrixForm

b4=KroneckerProduct[ψ1,ψ2, ψ2];b4//MatrixForm

b5=KroneckerProduct[ψ2,ψ1, ψ1]; b5//MatrixForm

b6=KroneckerProduct[ψ2,ψ1, ψ2]; b6//MatrixForm

b7=KroneckerProduct[ψ2,ψ2, ψ1]; b7//MatrixForm

b8=KroneckerProduct[ψ2,ψ2, ψ2]; b8//MatrixForm

It is confirmed that a11=a1, a22 = a2, a33 = a3, a44=a4, a55=a5, a66=a6,
 a77=a7, and  a88 = a8.

a11=UaH.b1//Simplify; a11//MatrixForm

a22=UaH.b2//Simplify;a22//MatrixForm

a33=UaH.b3//Simplify;a33//MatrixForm

a44=UaH.b4//Simplify; a44//MatrixForm

a55=UaH.b5//Simplify;a55//MatrixForm

a66=UaH.b6//Simplify; a66//MatrixForm

a77=UaH.b7//Simplify; a77//MatrixForm

a88=UaH.b8//Simplify; a88//MatrixForm

III.	Results from L.I. Schiff, Quantum Mechanics

L.I. Skiff, Quantum Mechanics, 3rd edition (McGraw-Hill, 1968)
p.377

Here we discuss the equivalence between our result  and  the expressions given by Schiff.
The following unitary operator is defined as Us (the index s comes from the name of  Schiff.

Us=(({{1, 0, 0, 0, 0, 0, 0, 0}, {0, (1/(Sqrt[3])), 0, 0, ((-1)/(Sqrt[6])), 0, (1/(Sqrt[2])), 0}, {0, (1/(Sqrt[3])), 0, 0, ((-1)/(Sqrt[6])), 0, ((-1)/(Sqrt[2])), 0}, {0, 0, (1/(Sqrt[3])), 0, 0, ((-2)/(Sqrt[6])), 0, 0}, {0, (1/(Sqrt[3])), 0, 0, (2/(Sqrt[6])), 0, 0, 0}, {0, 0, (1/(Sqrt[3])), 0, 0, (1/(Sqrt[6])), 0, (1/(Sqrt[2]))}, {0, 0, (1/(Sqrt[3])), 0, 0, (1/(Sqrt[6])), 0, ((-1)/(Sqrt[2]))}, {0, 0, 0, 1, 0, 0, 0, 0}}));

UsT=Transpose[Us], UsH=(SuperPlus[Us])
(SuperPlus[Us])Us=1

UsT=Transpose[Us];

UsH=Transpose[(SuperStar[Us])];

UsH.Us//MatrixForm

s1=UsT[[1]]; s1//MatrixForm

s2=UsT[[2]]; s2//MatrixForm

s3=UsT[[3]]; s3//MatrixForm

s4=UsT[[4]]; s4//MatrixForm

s5=UsT[[5]]; s5//MatrixForm

s6=UsT[[6]]; s6//MatrixForm

s7=UsT[[7]]; s7//MatrixForm

s8=UsT[[8]]; s8//MatrixForm

D3/2 (j = 3/2, s1, s2, s3, s4),
D1/2 ( j = 1/2, s5, s6)
D1/2 ( j = 1/2, s7, s8)

ST1.s1-(15/4) s1//Simplify

Sz.s1 - (3/2) s1

ST1.s2-(15/4)  s2//Simplify

Sz.s2-(1/2)s2//Simplify

ST1.s3-(15/4)  s3//Simplify

Sz.s3+(1/2)s3//Simplify

ST1.s4-(15/4)  s4//Simplify

Sz.s4+(3/2)s4//Simplify

D1/2 (j=1/2, m=1/2, -1/2)

ST1.s5-(3/4)  s5//Simplify

Sz.s5-(1/2)s5//Simplify

ST1.s6-(3/4)  s6//Simplify

Sz.s6+(1/2)s6//Simplify

D1/2 (j = 1/2, m = 1/2, -1/2)

ST1.s7-(3/4)  s7//Simplify

Sz.s7-(1/2)s7//Simplify

ST1.s8-(3/4)  s8//Simplify

Sz.s8+(1/2)s8//Simplify

|si>=Us|bi>,		|bi>=UsH|si>

UsH.s1//Simplify

UsH.s2//Simplify

UsH.s3//Simplify

UsH.s4//Simplify

UsH.s5//Simplify

UsH.s6//Simplify

UsH.s7//Simplify

UsH.s8//Simplify

UaH.Us//Simplify//MatrixForm

UsH.Ua//Simplify//MatrixForm

Det[Us]//Simplify

III.     Result from Tomonaga: 



Picture of Prof. Sin-itiro Tomonaga (1906-1979)

Sin-ItiroTomonaga 
Angular Momentum and Spin, Supplement to Quantum Mechanics I (1962) and II  (1966) (North-Holland), edited by Susumu Kamebuchi, Yasuo Hara, and Takeyasu Kodera 
(Misuzu-Shobou, 1988, in Japanese),	Page: 46



.

Here we discuss the equivalence between our result  and  the expressions given by TomonagaThe following unitary operator is defined as Ut (t comes from the name of  Tomonaga.

Ut=(({{1, 0, 0, 0, 0, 0, 0, 0}, {0, (1/(Sqrt[3])), 0, 0, ((-2)/(Sqrt[6])), 0, 0, 0}, {0, (1/(Sqrt[3])), 0, 0, (1/(Sqrt[6])), 0, (1/(Sqrt[2])), 0}, {0, 0, (1/(Sqrt[3])), 0, 0, (1/(Sqrt[6])), 0, ((-1)/(Sqrt[2]))}, {0, (1/(Sqrt[3])), 0, 0, (1/(Sqrt[6])), 0, ((-1)/(Sqrt[2])), 0}, {0, 0, (1/(Sqrt[3])), 0, 0, (1/(Sqrt[6])), 0, (1/(Sqrt[2]))}, {0, 0, (1/(Sqrt[3])), 0, 0, ((-2)/(Sqrt[6])), 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}}));

Det[Ut]

UtT=Transpose[Ut], 
UtH=(SuperPlus[Ut])
(SuperPlus[Ut])Ut=1

UtT=Transpose[Ut];

UtH=Transpose[(SuperStar[Ut])];

UtH.Ut//MatrixForm

t1=UtT[[1]]; t1//MatrixForm

t2=UtT[[2]]; t2//MatrixForm

t3=UtT[[3]]; t3//MatrixForm

t4=UtT[[4]]; t4//MatrixForm

t5=UtT[[5]]; t5//MatrixForm

t6=UtT[[6]]; t6//MatrixForm

t7=UtT[[7]]; t7//MatrixForm

t8=UtT[[8]]; t8//MatrixForm

D3/2 (j = 3/2, t1, t2, t3, t4),
D1/2 ( j = 1/2, t5, t6)
D1/2 ( j = 1/2, t7, t8)

ST1.t1-(15/4) t1//Simplify

Sz.t1 - (3/2) t1

ST1.t2-(15/4)  t2//Simplify

Sz.t2-(1/2)t2//Simplify

ST1.t3-(15/4)  t3//Simplify

Sz.t3+(1/2)t3//Simplify

ST1.t4-(15/4)  t4//Simplify

Sz.t4+(3/2)t4//Simplify

D1/2 (j=1/2, m=1/2, -1/2)

ST1.t5-(3/4)  t5//Simplify

Sz.t5-(1/2)t5//Simplify

ST1.t6-(3/4)  t6//Simplify

Sz.t6+(1/2)t6//Simplify

D1/2 (j = 1/2, m = 1/2, -1/2)

ST1.t7-(3/4)  t7//Simplify

Sz.t7-(1/2)t7//Simplify

ST1.t8-(3/4)  t8//Simplify

Sz.t8+(1/2)t8//Simplify

|ti>=Ut|bi>,		|bi>=UtH|ai>

UtH.t1//Simplify

UtH.t2//Simplify

UtH.t3//Simplify

UtH.t4//Simplify

UtH.t5//Simplify

UtH.t6//Simplify

UtH.t7//Simplify

UtH.t8//Simplify

UaH.Ut//Simplify//MatrixForm

UtH.Ua//Simplify//MatrixForm

Det[Ut]//Simplify

((Summary))