Elements of Modern Physics
370
One also imposes a phase condition for the determine of the matrix formed
by the coefficients x i and y i
x 1 y 2 – x 2 y 1 = 1
(10.7)
The linear transformations in Eq. (10.5), with the conditions in Eqs. (10.6)
and (10.7) define the group SU(2) (group of special unitary transformations in
2-dimensions) which is closely related to the usual 3-dimensional rotations.
Invariance under these transformations gives rise to SU(2) or isospin symmetry.
It allows the characterization of states by isospin I and its z-component. I z (similar
to l and m in the case of ordinary rotations). Thus, (u,d) have I = 1/2 and
I z = ± 1/2, (s) has I = 0 and I z = 0, (P, N) have I = 1/2 and I z = ± 1/2, (Σ
+
, Σ
0
, Σ
–
)
have I = 1 and I z = 1, 0, –1, etc. Furthermore, these quantum numbers are
conserved in processes which involve only strong interaction.
For specific applications, the (P, N) system is considered which can have
I = 1 or 0. Designating the isospin states by | I, I z 〉,
| 1,1〉 = | PP〉
(10.8)
| 1, 0〉 = 1/2
1
2
(| PN 〉 + | NP 〉)
(10.9)
| 1, –1〉 = | NN〉
(10.10)
| 0, 0〉 = 1/2
1
2
(| PN〉 – | NP〉)
(10.11)
These relations follow from the usual quantum-mechanical rules for
combining two angular momenta (also see Problem 1). Isospin symmetry then
implies that the probability amplitudes T, which are essentially the probability
amplitudes for the processes, satisfy the relations
〈PP | T | PP〉 = 〈 NN| T |NN〉
=
1
2
〈PN + NP | T | PN + NP〉
(10.12)
Another useful application is obtained by noting that the deuteron D appears
in only one charge state and hence is assigned I = 0. Since the π
–
meson multiplet
has I = 1, the Dπ state is an I = 1 state. Conservation of isospin then gives the
result
〈 Dπ
0
| T | PN〉 = 1/2
1
2
〈Dπ
+
| T | PP〉
(10.13)
Experimentally, this relation was verified at an energy of 340 MeV, to with
in a few per cent by Hildebrand (1953), which supports the general ideas of
isospin invariance in strong interaction.
SU(3) and Higher Symmetries
If the masses of the baryons in Table 10.2 are examined, it is observed that even
baryons with different I have approximately equal masses (the difference are
370
One also imposes a phase condition for the determine of the matrix formed
by the coefficients x i and y i
x 1 y 2 – x 2 y 1 = 1
(10.7)
The linear transformations in Eq. (10.5), with the conditions in Eqs. (10.6)
and (10.7) define the group SU(2) (group of special unitary transformations in
2-dimensions) which is closely related to the usual 3-dimensional rotations.
Invariance under these transformations gives rise to SU(2) or isospin symmetry.
It allows the characterization of states by isospin I and its z-component. I z (similar
to l and m in the case of ordinary rotations). Thus, (u,d) have I = 1/2 and
I z = ± 1/2, (s) has I = 0 and I z = 0, (P, N) have I = 1/2 and I z = ± 1/2, (Σ
+
, Σ
0
, Σ
–
)
have I = 1 and I z = 1, 0, –1, etc. Furthermore, these quantum numbers are
conserved in processes which involve only strong interaction.
For specific applications, the (P, N) system is considered which can have
I = 1 or 0. Designating the isospin states by | I, I z 〉,
| 1,1〉 = | PP〉
(10.8)
| 1, 0〉 = 1/2
1
2
(| PN 〉 + | NP 〉)
(10.9)
| 1, –1〉 = | NN〉
(10.10)
| 0, 0〉 = 1/2
1
2
(| PN〉 – | NP〉)
(10.11)
These relations follow from the usual quantum-mechanical rules for
combining two angular momenta (also see Problem 1). Isospin symmetry then
implies that the probability amplitudes T, which are essentially the probability
amplitudes for the processes, satisfy the relations
〈PP | T | PP〉 = 〈 NN| T |NN〉
=
1
2
〈PN + NP | T | PN + NP〉
(10.12)
Another useful application is obtained by noting that the deuteron D appears
in only one charge state and hence is assigned I = 0. Since the π
–
meson multiplet
has I = 1, the Dπ state is an I = 1 state. Conservation of isospin then gives the
result
〈 Dπ
0
| T | PN〉 = 1/2
1
2
〈Dπ
+
| T | PP〉
(10.13)
Experimentally, this relation was verified at an energy of 340 MeV, to with
in a few per cent by Hildebrand (1953), which supports the general ideas of
isospin invariance in strong interaction.
SU(3) and Higher Symmetries
If the masses of the baryons in Table 10.2 are examined, it is observed that even
baryons with different I have approximately equal masses (the difference are