174
7 Spherical Symmetry and Spins
Table 7.2 The coupling coefficients for the direct product of two spins
S = 0
S = 1
|+1| 0| − 1| x| y| z
|α|α
01
0
0
−1/
√
2
i/
√
20
|α|β
1/
√
20
1 /
√
20
0
0
1 /
√
2
|β|α− 1/
√
20
1 /
√
20
0
0
1 /
√
2
|β|β
00
0
1
1 /
√
2
i/
√
20
interchange corresponds to the following matrix transformation:
|α| β
0 −1
10
(7.37)
This matrix gives rise to the mapping of U onto its complex conjugate:
0 −1
10
ab
− ¯
b ¯
a
01
−10
=
¯
a
¯
b
−ba
(7.38)
For this reason, the matrix in Eq. (7.37) is also called the conjugating matrix.The
conjugating matrix is defined up to an arbitrary phase. We have taken here the standard phase choice. The conjugating relationships between the two spins can now be
used to transfer the spin functions in the coupling coefficients from the ket to the
bra part. An α spin in the ket part becomes a β spin in the bra part, while a β spin
in the ket becomes a −α spin in the bra part.
The corresponding coupling coefficients are summarized in Table 7.2. They indicate how two spins can be coupled to a vector. Here, we also express the vector in
complex form, as
|+1=−
1
√
2
|x+i|y
|0=|z
(7.39)
|−1=
1
√
2
|x−i|y
This transformation is in accordance with the Condon–Shortley phase conventions
for the spherical basis functions [7]. In fact, our initial Hamiltonian matrix in
Eq. (7.21) was constructed in this way. The resulting vector corresponds to the triplet
spin functions, which we used in Sect. 6.4. The total spinor product space has dimension 4. The remainder after extraction of the three triplet functions corresponds
to the spin singlet, which is invariant and transforms as a scalar. Spinors are thus
the fundamental building blocks of 3D space. Their transformation properties were
known to Rodrigues as early as 1840. It was some ninety years before Pauli realized
that elementary particles, such as electrons, had properties that could be described
7 Spherical Symmetry and Spins
Table 7.2 The coupling coefficients for the direct product of two spins
S = 0
S = 1
|+1| 0| − 1| x| y| z
|α|α
01
0
0
−1/
√
2
i/
√
20
|α|β
1/
√
20
1 /
√
20
0
0
1 /
√
2
|β|α− 1/
√
20
1 /
√
20
0
0
1 /
√
2
|β|β
00
0
1
1 /
√
2
i/
√
20
interchange corresponds to the following matrix transformation:
|α| β
0 −1
10
(7.37)
This matrix gives rise to the mapping of U onto its complex conjugate:
0 −1
10
ab
− ¯
b ¯
a
01
−10
=
¯
a
¯
b
−ba
(7.38)
For this reason, the matrix in Eq. (7.37) is also called the conjugating matrix.The
conjugating matrix is defined up to an arbitrary phase. We have taken here the standard phase choice. The conjugating relationships between the two spins can now be
used to transfer the spin functions in the coupling coefficients from the ket to the
bra part. An α spin in the ket part becomes a β spin in the bra part, while a β spin
in the ket becomes a −α spin in the bra part.
The corresponding coupling coefficients are summarized in Table 7.2. They indicate how two spins can be coupled to a vector. Here, we also express the vector in
complex form, as
|+1=−
1
√
2
|x+i|y
|0=|z
(7.39)
|−1=
1
√
2
|x−i|y
This transformation is in accordance with the Condon–Shortley phase conventions
for the spherical basis functions [7]. In fact, our initial Hamiltonian matrix in
Eq. (7.21) was constructed in this way. The resulting vector corresponds to the triplet
spin functions, which we used in Sect. 6.4. The total spinor product space has dimension 4. The remainder after extraction of the three triplet functions corresponds
to the spin singlet, which is invariant and transforms as a scalar. Spinors are thus
the fundamental building blocks of 3D space. Their transformation properties were
known to Rodrigues as early as 1840. It was some ninety years before Pauli realized
that elementary particles, such as electrons, had properties that could be described