Addition of the spin angular momentum of three (four...
30
1 Addition of the spin angular momentum of three (four) electrons with S = 1/2 Masatsugu Suzuki Department of Physics, SUNY at Binghamton (Date: October 23, 2013) Here we show how to derive the eigenvalues and eigenkets of 2 ˆ S , the spin states of the three and four electrons with spin 1/2, where S ˆ is the total spin angular momentum defined by 3 2 1 ˆ ˆ ˆ ˆ S S S S , for the three particles 4 3 2 1 ˆ ˆ ˆ ˆ ˆ S S S S S . for the four particles There are two methods. One is the conventional method to use the Clebsch-Gordan coefficients for the addition of spin angular momentum. Using the program of ClebschGordan (Mathematica), we can derive the expression for the eigenket and eigenvalues for 2 ˆ S for the many spin particles. The second method is to use the KroneckerProduct of the Mathematica for both the three and four spins. We calculate the expression of the matrix for 2 ˆ S ; 8x8 matrix for the three spin particles and 16x16 matrix for the four spin particles. Using the Mathematica we solve the eigenvalue problems to get the eigenvalues and eigenkets for each case. We can also solve the eigenvalue problem even for the five spin particles, the 32x32 martix by using the Mathematica. Note that for the n particle systems, we need to solve the eigenvalue problem for 2 n x2 n martix (n = 2, 3, 4, 5, 6, ...). 1. States of three particles with spin 1/2 (I) We can regard three electrons as 2+1 electrons, in the sense that we can combine an electron (s = 1/2) with the triplet two-electron state (s = 1) and with the singlet two-electron state (s = 0). 0 1 2 / 1 2 / 1 D D D D . ((Case-1)) 2 / 1 2 / 1 2 / 3 2 / 1 0 2 / 1 1 2 / 1 0 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 ) ( ) ( D D D D D D D D D D D D D D D D In the case of ) ( 2 / 1 2 / 3 D D , the results on the addition of the angular momenta shows that we should get two groups of three-electron spin states corresponding to S = 3/2 and S = 1/2. In the
Transcript of Addition of the spin angular momentum of three (four...
1
Addition of the spin angular momentum of three (four) electrons with S = 1/2 Masatsugu Suzuki
Department of Physics, SUNY at Binghamton (Date: October 23, 2013)
Here we show how to derive the eigenvalues and eigenkets of 2S , the spin states of the three and
four electrons with spin 1/2, where S is the total spin angular momentum defined by
321ˆˆˆˆ SSSS , for the three particles
4321ˆˆˆˆˆ SSSSS . for the four particles
There are two methods. One is the conventional method to use the Clebsch-Gordan coefficients for the addition of spin angular momentum. Using the program of ClebschGordan (Mathematica),
we can derive the expression for the eigenket and eigenvalues for 2S for the many spin particles. The second method is to use the KroneckerProduct of the Mathematica for both the three and
four spins. We calculate the expression of the matrix for 2S ; 8x8 matrix for the three spin particles and 16x16 matrix for the four spin particles. Using the Mathematica we solve the eigenvalue problems to get the eigenvalues and eigenkets for each case. We can also solve the eigenvalue problem even for the five spin particles, the 32x32 martix by using the Mathematica.
Note that for the n particle systems, we need to solve the eigenvalue problem for 2nx2n martix (n = 2, 3, 4, 5, 6, ...). 1. States of three particles with spin 1/2 (I)
We can regard three electrons as 2+1 electrons, in the sense that we can combine an electron (s = 1/2) with the triplet two-electron state (s = 1) and with the singlet two-electron state (s = 0).
012/12/1 DDDD .
((Case-1))
2/12/12/3
2/102/11
2/101
2/12/12/12/12/12/1
)(
)(
DDD
DDDD
DDD
DDDDDD
In the case of )( 2/12/3 DD , the results on the addition of the angular momenta shows that we
should get two groups of three-electron spin states corresponding to S = 3/2 and S = 1/2. In the
2
second case we get a single group that corresponds to S = 1/2. We thus expect one quartet group of spin states (S = 3/2) and two distinct doublet groups of spin states (S = 1/2), or a total of 4+2+2 = 8 individual three-electron spin states. (1) j = 3/2.
)3()2()1(2
3,
2
3 ,
3
)3()2()1()3()2()1()3()2()1(
2
1,
2
3
3
)3()2()1()3()2()1()3()2()1(
2
1,
2
3
)3()2()1(2
3,
2
3
___________________________________________________________________________ (ii) j = 1/2.
6
)3()2()1(2)3()2()1()3()2()1(
2
1,
2
1
6
)3()2()1()3()2()1(2)3()2()1(
2
1,
2
1
_____________________________________________________________________________ (iii) j = 1/2
2
)3()2()1()3()2()1(
2
1,
2
1
2
)3()2()1()3()2()1(
2
1,
2
1
Note that and denote the upper spin state and the lower spin state, respectively;
3
0
1z ,
0
1z
The numbers 1, 2, and 3 are the sites of spins. These results are the same as those derived by Tomonaga, except for the sign. ((Mathematica-1))
Clear"Global`"; CCGGj1_, m1_, j2_, m2_, j_, m_ :
Modules1,
s1 IfAbsm1 b j1 && Absm2 b j2 && Absm b j,
ClebschGordanj1, m1, j2, m2, j, m, 0;
CG2j1_, j2_, j_, a1_, a2_ :
TableSumCCGGj1, k1, j2, k2, j, k1 k2 a1j1, k1 a2j2, k2KroneckerDeltak1 k2, m, k1, j1, j1, k2, j2, j2, m, j, j
j1=1/2, j2=1/2 j = 1, 0
CG212, 12, 1, b1, b2 TableForm
b1 12
, 12 b2 1
2, 1
2
b112
, 12 b2 1
2, 1
2
2
b112
, 12 b2 1
2, 1
2
2
b1 12
, 12 b2 1
2, 1
2
CG212, 12, 0, b1, b2 TableForm
b112
, 12 b2 1
2, 1
2
2
b112
, 12 b2 1
2, 1
2
2j1=1, j2=1/2 j = 3/2
j1 1; j2 12; j 32;
TableSumCCGGj1, k1, j2, k2, j, k1 k2 CG212, 12, j1, b1, b2k1 j1 1b3j2, k2 KroneckerDeltak1 k2, m, k1, j1, j1, k2, j2, j2, m, j, j
Simplify
b11
2,
1
2 b21
2,
1
2 b31
2,
1
2,
b1 12
, 12 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
3,
b1 12
, 12 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
3,
b11
2,
1
2 b2 1
2,
1
2 b3 1
2,
1
2
4
2. States of three particles with spin 1/2 (II)
We consider the second case. The results on the spin states are the same as those described in the textbook of Schiff.
2/12/12/3
02/112/1
012/1
2/12/12/12/12/12/1
)(
)(
DDD
DDDD
DDD
DDDDDD
(1) j = 3/2.
)3()2()1(2
3,
2
3 ,
3
)3()2()1()3()2()1()3()2()1(
2
1,
2
3 ,
3
)3()2()1()3()2()1()3()2()1(
2
1,
2
3 ,
j1=1, j2=1/2 j = 1/2
j1 1; j2 12; j 12;
TableSumCCGGj1, k1, j2, k2, j, k1 k2 CG212, 12, j1, b1, b2k1 j1 1b3j2, k2 KroneckerDeltak1 k2, m, k1, j1, j1, k2, j2, j2, m, j, j
Simplify
b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 2 b2 1
2, 1
2 b3 1
2, 1
2
6,
b1 12
, 12 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
6
j1=0, j2=1/2 j = 1/2
j1 0; j2 12; j 12;
TableSumCCGGj1, k1, j2, k2, j, k1 k2 CG212, 12, j1, b1, b2k1 j1 1b3j2, k2 KroneckerDeltak1 k2, m, k1, j1, j1, k2, j2, j2, m, j, j
Simplify
b1 1
2, 1
2 b2 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
2,
b1 12
, 12 b2 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
2
5
)3()2()1(2
3,
2
3 .
___________________________________________________________________________ (ii) j = 1/2.
6
)3()2()1()3()2()1()3()2()1(2
2
1,
2
1 ,
6
)3()2()1()3()2()1()3()2()1(2
2
1,
2
1 .
_____________________________________________________________________________ (iii) j = 1/2
2
)3()2()1()3()2()1(
2
1,
2
1 ,
2
)3()2()1()3()2()1(
2
1,
2
1 .
((Mathematica-2))
6
Clear"Global`"; CCGGj1_, m1_, j2_, m2_, j_, m_ :
Modules1,
s1 IfAbsm1 b j1 && Absm2 b j2 && Absm b j,
ClebschGordanj1, m1, j2, m2, j, m, 0;
CG2j1_, j2_, j_, a1_, a2_ :
TableSumCCGGj1, k1, j2, k2, j, k1 k2 a1j1, k1 a2j2, k2KroneckerDeltak1 k2, m, k1, j1, j1, k2, j2, j2, m, j, j
j1=1/2, j2=1/2 j = 1, 0
CG212, 12, 1, b1, b2 TableForm
b1 12
, 12 b2 1
2, 1
2
b112
, 12 b2 1
2, 1
2
2
b112
, 12 b2 1
2, 1
2
2
b1 12
, 12 b2 1
2, 1
2
CG212, 12, 0, b1, b2 TableForm
b112
, 12 b2 1
2, 1
2
2
b112
, 12 b2 1
2, 1
2
2j1=1, j2=1/2 j = 3/2
j1 1; j2 12; j 32;
TableSumb1j2, k2 CCGGj1, k1, j2, k2, j, k1 k2CG212, 12, j1, b2, b3k1 j1 1 KroneckerDeltak1 k2, m,
k1, j1, j1, k2, j2, j2, m, j, j Simplify
b11
2,
1
2 b21
2,
1
2 b31
2,
1
2,
b1 12
, 12 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
3,
b1 12
, 12 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
3,
b11
2,
1
2 b2 1
2,
1
2 b3 1
2,
1
2
j1=1, j2=1/2 j = 1/2
j1 1; j2 12; j 12;
TableSumb1j2, k2 CCGGj1, k1, j2, k2, j, k1 k2CG212, 12, j1, b2, b3k1 j1 1 KroneckerDeltak1 k2, m,
k1, j1, j1, k2, j2, j2, m, j, j Simplify
7
3. The three spin states (by the use of KroneckerProduct)
The magnitude of the total spin angular momentum:
)ˆˆˆˆˆˆ(2
14
9
)ˆˆ2ˆˆ2ˆˆ2ˆˆˆ(4
)ˆˆˆ(4
ˆ
133221
22
1332212
32
22
1
2
2321
22
σσσσσσ
σσσσσσσσσ
σσσS
and the z-component of the total spin angular momentum:
)ˆˆˆ(2
ˆ221 zzzzS
.
Using thr Kronecker product, the above operators can be rewritten as
)ˆˆ1ˆˆ1ˆˆ1
ˆ1ˆˆ1ˆˆ1ˆ
1ˆˆ1ˆˆ1ˆˆ(2
1114
9ˆ22
2
zzyyxx
zzyyxx
zzyyxx
S
)ˆ111ˆ111ˆ(2
ˆzzzzS
.
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
6,
2 b1 12
, 12 b2 1
2, 1
2 b3 1
2, 1
2 b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
6
j1=0, j2=1/2 j = 1/2
j1 0; j2 12; j 12;
TableSumb1j2, k2 CCGGj1, k1, j2, k2, j, k1 k2CG212, 12, j1, b2, b3k1 j1 1 KroneckerDeltak1 k2, m,
k1, j1, j1, k2, j2, j2, m, j, j Simplify
b1 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
2,
b1 12
, 12 b2 1
2, 1
2 b3 1
2, 1
2 b2 1
2, 1
2 b3 1
2, 1
2
2
8
The matrix of 2S is obtained with the use of KrocknerProduct in the Mathematica, as
The matrix of zS is obtained as
Note that zS is the block-diagonnal matrix,but 2S is a non-diagonal matrix.
The eigenvalue problem of 2S . We use the Eigensystem (Mathematica) to solve the eigenvalue problem. The result is as follows: {eigenvalues, eigenkets}}
9
The eigenkets of 2S (a)
0
0
0
0
0
0
0
1
2
3,
3
2mj ,
0
0
03
103
13
10
2
1,
2
3mj
03
13
103
10
0
0
2
1,
2
3mj ,
1
0
0
0
0
0
0
0
2
3,
2
3mj ,
(b) j = 1/2
154
,154
,154
,154
,34
,34
,34
,34,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0
10
02
10
02
10
0
0
2
1,
2
1mj ,
0
0
02
10
02
10
2
1,
2
1mj
(c) j = 1/2
0
0
06
10
6
26
10
2
1,
2
1mj ,
06
16
206
10
0
0
2
1,
2
1mj
4. The four-spin states (by the use of KroneckerProduct) We now consider the eigenstates and eigenvalues for the four-spin system.
012
011012
00100111
0101
2/12/12/12/12/12/12/12/1
23
)(
)()(
)()(
DDD
DDDDDD
DDDDDDDD
DDDD
DDDDDDDD
The total states are 16 states (= 5 + 3 x 3+2), since
j = 2 m = 2, 1, 0, -1, -2 (one) 1 x 5 = 5 j = 1 m = 1, 0, -1 (three) 3 x 3 = 9 j = 0 m = 0. (two) 2 x 2 = 4
11
The square of the magnitude of the total spin angular momentum
)ˆˆˆˆˆˆ
ˆˆˆˆˆˆ(2
13
)ˆˆ2ˆˆ2ˆˆ2
ˆˆ2ˆˆ2ˆˆ2ˆˆˆˆ(4
)ˆˆˆˆ(4
ˆ
434241
323121
22
434241
3231212
42
32
22
1
2
24321
22
σσσσσσ
σσσσσσ
σσσσσσ
σσσσσσσσσσ
σσσσS
The z component of the spin angular momentum:
)ˆˆˆˆ(2
ˆ4321 zzzzzS
These operators can be rewritten as
)ˆˆ11ˆˆ11ˆˆ11
ˆ1ˆ1ˆ1ˆ1ˆ1ˆ1
ˆ11ˆˆ11ˆˆ11ˆ
1ˆˆ11ˆˆ11ˆˆ1
1ˆ1ˆ1ˆ1ˆ1ˆ1ˆ
11ˆˆ11ˆˆ11ˆˆ(2
11113ˆ2
22
zzyyxx
zzyyxx
zzyyxx
zzyyxx
zzyyxx
zzyyxx
S
and
)ˆ111ˆ111ˆ(2
ˆzzzzS
Using the Mathematica we can solve the eigenvalue problems for the four spin 1/2 systems. The
matrix of 2S (16 x 16 matrix) is expressed by
12
The matrix of zS (16 x 16 matrix) is expressed as
The eigenvalue problem of 2S (a) j = 2
13
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2,2 mj
,
0
0
0
0
0
0
0
2/1
0
0
0
2/1
0
2/1
2/1
0
1,2 mj
0
0
0
6/1
0
6/1
6/1
0
0
6/1
6/1
0
6/1
0
0
0
0,2 mj
0
2/1
2/1
0
2/1
0
0
0
2/1
0
0
0
0
0
0
0
1,2 mj
14
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2,2 mj
_____________________________________________________________________________ (b)
0
0
0
0
0
0
0
2/1
0
0
0
0
0
0
2/1
0
1,1 mj
0
0
0
0
0
0
0
6/1
0
0
0
6/2
0
0
6/1
0
1,1 mj
0
0
0
0
0
0
0
6/3
0
0
0
6/3
0
2/3
6/3
0
1,1 mj
15
0
0
0
2/1
0
0
0
0
0
0
0
0
2/1
0
0
0
0,1 mj
,
0
0
0
0
0
2/1
0
0
0
0
2/1
0
0
0
0
0
0,1 mj
0
0
0
0
0
0
2/1
0
0
2/1
0
0
0
0
0
0
0,1 mj
0
2/1
0
0
0
0
0
0
2/1
0
0
0
0
0
0
0
1,1 mj
0
6/1
6/2
0
0
0
0
0
6/1
0
0
0
0
0
0
0
1,1 mj
0
6/3
6/3
0
2/3
0
0
0
6/3
0
0
0
0
0
0
0
1,1 mj
______________________________________________________________________________
16
0
0
0
2/1
0
0
2/1
0
0
2/1
0
0
2/1
0
0
0
0,0 mj
___________________________________________________________________________ (
0
0
0
6/3
0
3/1
6/3
0
0
6/3
3/1
0
6/3
0
0
0
0,0 mj
____________________________________________________________________________
17
((Mathematica))
18
Clear"Global`";
exp_ : exp . Complexre_, im_ Complexre, im; x 0 11 0
;
y 0 0
; z 1 00 1
;
I2 IdentityMatrix2;
ST1 1
2KroneckerProductx, x, I2, I2 KroneckerProducty, y, I2, I2 KroneckerProductz, z, I2, I2 KroneckerProductx, I2, x, I2 KroneckerProducty, I2, y, I2 KroneckerProductz, I2, z, I2 KroneckerProductx, I2, I2, x KroneckerProducty, I2, I2, y KroneckerProductz, I2, I2, z KroneckerProductI2, x, x, I2 KroneckerProductI2, y, y, I2 KroneckerProductI2, z, z, I2 KroneckerProductI2, I2, x, x KroneckerProductI2, I2, y, y KroneckerProductI2, I2, z, z KroneckerProductI2, x, I2, x KroneckerProduct I2, y, I2, y KroneckerProductI2, z, I2, z
3 KroneckerProductI2, I2, I2, I2;
ST1 MatrixForm
6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 3 1 0 1 0 0 0 1 0 0 0 0 0 0 00 1 3 0 1 0 0 0 1 0 0 0 0 0 0 00 0 0 2 0 1 1 0 0 1 1 0 0 0 0 00 1 1 0 3 0 0 0 1 0 0 0 0 0 0 00 0 0 1 0 2 1 0 0 1 0 0 1 0 0 00 0 0 1 0 1 2 0 0 0 1 0 1 0 0 00 0 0 0 0 0 0 3 0 0 0 1 0 1 1 00 1 1 0 1 0 0 0 3 0 0 0 0 0 0 00 0 0 1 0 1 0 0 0 2 1 0 1 0 0 00 0 0 1 0 0 1 0 0 1 2 0 1 0 0 00 0 0 0 0 0 0 1 0 0 0 3 0 1 1 00 0 0 0 0 1 1 0 0 1 1 0 2 0 0 00 0 0 0 0 0 0 1 0 0 0 1 0 3 1 00 0 0 0 0 0 0 1 0 0 0 1 0 1 3 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6
19
eq1 EigensystemST1 Simplify
6, 6, 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0,
0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0,
0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0
20
1 eq12, 1; 2 eq12, 2; 3 eq12, 3; 4 eq12, 4;
5 eq12, 5; 6 eq12, 6; 7 eq12, 7; 8 eq12, 8;
9 eq12, 9; 10 eq12, 10; 11 eq12, 11; 12 eq12, 12;
13 eq12, 13; 14 eq12, 14; 15 eq12, 15;
16 eq12, 16;
eq2 Orthogonalize1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0,12
, 0, 0, 0,12
, 0,12
,12
, 0,
0, 0, 0,1
6, 0,
1
6,
1
6, 0, 0,
1
6,
1
6, 0,
1
6, 0, 0, 0,
0,12
,12
, 0,12
, 0, 0, 0,12
, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1
2, 0, 0, 0, 0, 0, 0,
1
2, 0,
0, 0, 0, 0, 0, 0, 0, 1
6, 0, 0, 0, 0, 0,
23
, 1
6, 0,
0, 0, 0, 1
2, 0, 0, 0, 0, 0, 0, 0, 0,
1
2, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1
2 3, 0, 0, 0,
32
, 0, 1
2 3,
1
2 3, 0,
0, 0, 0, 0, 0, 1
2, 0, 0, 0, 0,
1
2, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1
2, 0, 0,
1
2, 0, 0, 0, 0, 0, 0,
0, 1
2, 0, 0, 0, 0, 0, 0,
1
2, 0, 0, 0, 0, 0, 0, 0,
0, 1
6, 0, 0,
23
, 0, 0, 0, 1
6, 0, 0, 0, 0, 0, 0, 0,
0, 1
2 3,
32
, 0, 1
2 3, 0, 0, 0,
1
2 3, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0,12
, 0, 0, 12
, 0, 0, 12
, 0, 0,12
, 0, 0, 0,
0, 0, 0, 1
2 3, 0,
1
3,
1
2 3, 0, 0,
1
2 3,
1
3, 0,
1
2 3, 0, 0, 0
21
1 eq24; 2 eq23; 3 eq22; 4 eq21; 5 eq25;
6 eq26; 7 eq27; 8 eq28; 9 eq29; 10 eq210;
11 eq211; 12 eq212; 13 eq213; 14 eq214;
15 eq215;
16 eq216;
Sz 1
2KroneckerProductz, I2, I2, I2 KroneckerProductI2, z, I2, I2
KroneckerProductI2, I2, z, I2 KroneckerProductI2, I2, I2, z;
Sz MatrixForm
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2
22
ST1.1 6 1
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.1 1
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.2 6 2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.3 6 3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.3 3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
23
ST1.4 6 4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.4 2 4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.5 6 5
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.5 2 5
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.6 2 6
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.6 6
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.7 2 7 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
24
Sz.7 7
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.8 2 8 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.8
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.9 2 9 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.9 9
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.10 2 10 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.10
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
25
ST1.11 2 11 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.11
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.12 2 12 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.12 12
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.13 2 13 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.13 13
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.14 2 14 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
26
Sz.14 14
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.15 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.15
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
ST1.16 Simplify
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Sz.16
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4.5, 5.6, 6.7, 7.80, 0, 0, 01
0,12
,12
, 0,12
, 0, 0, 0,12
, 0, 0, 0, 0, 0, 0, 0
2
0, 0, 0,1
6, 0,
1
6,
1
6, 0, 0,
1
6,
1
6, 0,
1
6, 0, 0, 0
3
0, 0, 0, 0, 0, 0, 0,12
, 0, 0, 0,12
, 0,12
,12
, 0
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
5
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
27
6
0, 0, 0, 0, 0, 0, 0, 1
2, 0, 0, 0, 0, 0, 0,
1
2, 0
7
0, 0, 0, 0, 0, 0, 0, 1
6, 0, 0, 0, 0, 0,
23
, 1
6, 0
8
0, 0, 0, 1
2, 0, 0, 0, 0, 0, 0, 0, 0,
1
2, 0, 0, 0
9
0, 0, 0, 0, 0, 0, 0, 1
2 3, 0, 0, 0,
32
, 0, 1
2 3,
1
2 3, 0
10
0, 0, 0, 0, 0, 1
2, 0, 0, 0, 0,
1
2, 0, 0, 0, 0, 0
28
____________________________________________________________________________ REFERENCES L.I. Schiff, Quantum Mechanics third edition (McGraw-Hill Book Company, New York, 1968). S. Tomonaga, Angylar momentum and spin (Misuzu, Toyo, 1989). D.M. Brink and G.R. Satcher, Angular Momentum, second edition (Clarendon Press, Oxford,
1966). Nouredine Zettili, Quantum Mechanics, Concepts and Applications, 2nd edition (John Wiley &
Sons, New York, 2009).
_____________________________________________________________________________ APPENDIX Mathematica 1. Clebsch-Gordan coefficient
11
0, 0, 0, 0, 0, 0, 1
2, 0, 0,
1
2, 0, 0, 0, 0, 0, 0
12
0, 1
2, 0, 0, 0, 0, 0, 0,
1
2, 0, 0, 0, 0, 0, 0, 0
13
0, 1
6, 0, 0,
23
, 0, 0, 0, 1
6, 0, 0, 0, 0, 0, 0, 0
14
0, 1
2 3,
32
, 0, 1
2 3, 0, 0, 0,
1
2 3, 0, 0, 0, 0, 0, 0, 0
15
0, 0, 0,12
, 0, 0, 12
, 0, 0, 12
, 0, 0,12
, 0, 0, 0
16
0, 0, 0, 1
2 3, 0,
1
3,
1
2 3, 0, 0,
1
2 3,
1
3, 0,
1
2 3, 0, 0, 0
29
2. KroneckerProduct
3. Eigensystem
30
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