7.4 The Coupling of Spins
173
Fig. 7.3 Homomorphism
between SU(2) and SO(3)
ˆ
R(−α, −n) leaves the Cayley–Klein parameters unchanged. By contrast, the combinations ˆ
R(2π − α, −n) and ˆ
R(−2π + α, n) change the signs of both Cayley–Klein
parameters.
7.4 The Coupling of Spins
The entries in the Hamiltonian provide us with coupling coefficients between vector and spinor. Applying the Wigner–Eckart theorem to the matrix elements of the
Zeeman spin Hamiltonian and separating out the constant parameters for the magnetic field yield the following nonzero coupling coefficients in the spin operator S,
where K is the reduced matrix element:
α| ˆ
S z |α=Kα|zα=1/2
β| ˆ
S z |β=Kβ|zβ =−1/2
α| ˆ
S x |β=Kα|xβ=1/2
β| ˆ
S x |α=Kβ|xα=1/2
α| ˆ
S y |β=Kα|yβ=−i/2
β| ˆ
S y |α=Kβ|zα=i/2
(7.36)
These coefficients can also be reversed to describe the coupling between two spinors
to yield a vector. This requires that the ket spin in the coefficients is relocated to
the bra part. In Sect. 6.3 we indicated that, for real functions, such a shift does
not change the coupling, except for a renormalization factor. The transposition of
the spin functions is more delicate since the bra functions transform as the complex conjugate of the ket functions, as indicated in Eq. (7.28). In order to establish
the equivalences between transformations of bra and ket spins, one must identify
the basis transformation that turns the matrix U into its complex conjugate. This
transformation is readily achieved by replacing |α by |β, and |β by −|α.T h e
173
Fig. 7.3 Homomorphism
between SU(2) and SO(3)
ˆ
R(−α, −n) leaves the Cayley–Klein parameters unchanged. By contrast, the combinations ˆ
R(2π − α, −n) and ˆ
R(−2π + α, n) change the signs of both Cayley–Klein
parameters.
7.4 The Coupling of Spins
The entries in the Hamiltonian provide us with coupling coefficients between vector and spinor. Applying the Wigner–Eckart theorem to the matrix elements of the
Zeeman spin Hamiltonian and separating out the constant parameters for the magnetic field yield the following nonzero coupling coefficients in the spin operator S,
where K is the reduced matrix element:
α| ˆ
S z |α=Kα|zα=1/2
β| ˆ
S z |β=Kβ|zβ =−1/2
α| ˆ
S x |β=Kα|xβ=1/2
β| ˆ
S x |α=Kβ|xα=1/2
α| ˆ
S y |β=Kα|yβ=−i/2
β| ˆ
S y |α=Kβ|zα=i/2
(7.36)
These coefficients can also be reversed to describe the coupling between two spinors
to yield a vector. This requires that the ket spin in the coefficients is relocated to
the bra part. In Sect. 6.3 we indicated that, for real functions, such a shift does
not change the coupling, except for a renormalization factor. The transposition of
the spin functions is more delicate since the bra functions transform as the complex conjugate of the ket functions, as indicated in Eq. (7.28). In order to establish
the equivalences between transformations of bra and ket spins, one must identify
the basis transformation that turns the matrix U into its complex conjugate. This
transformation is readily achieved by replacing |α by |β, and |β by −|α.T h e