If the nucleons are in a state of relative orbital angular...

If the nucleons are in a state of relative orbital angular momentum $L=0$, use the Pauli exclusion principle to show that $S+I$ must be an odd integer. There is much evidence to show that the nuclear force is invariant under isospin transformations [for example, that it is independent of the value of $I_{3}$ in the $I=1$ multiplet of $(2.2)$. For instance, consider the three nuclei ${ }^{6} \mathrm{He},{ }^{6} \mathrm{Li}$, and ${ }^{6} \mathrm{Be}$, which can be regarded respectively as an $\mathrm{nn}, \mathrm{np}$, and pp system attached to ${ }^{4} \mathrm{He}$ core of $I=0$. After correcting for the Coulomb repulsion between the protons and for the neutron-proton mass difference, the observed nuclear masses are as sketched in Fig. 2.2. Furthermore, isospin invariance requires that the same nuclear physics should be obtained for each of the three $I=1$ states $\left(I_{3}=\right.$ $-1,0,1)$, just as rotational invariance ensures that the $2 J+1$ substates of an isolated system of total angular momentum $J$ describe exactly equivalent physical systems.

If the nucleons are in a state of relative orbital angular momentum $L=0$, use the Pauli exclusion principle to show that $S+I$ must be an odd integer.
There is much evidence to show that the nuclear force is invariant under isospin transformations [for example, that it is independent of the value of $I_{3}$ in the $I=1$ multiplet of $(2.2)$. For instance, consider the three nuclei ${ }^{6} \mathrm{He},{ }^{6} \mathrm{Li}$, and ${ }^{6} \mathrm{Be}$, which can be regarded respectively as an $\mathrm{nn}, \mathrm{np}$, and pp system attached to ${ }^{4} \mathrm{He}$ core of $I=0$. After correcting for the Coulomb repulsion between the protons and for the neutron-proton mass difference, the observed nuclear masses are as sketched in Fig. 2.2. Furthermore, isospin invariance requires that the same nuclear physics should be obtained for each of the three $I=1$ states $\left(I_{3}=\right.$ $-1,0,1)$, just as rotational invariance ensures that the $2 J+1$ substates of an isolated system of total angular momentum $J$ describe exactly equivalent physical systems.

Key Concepts

Spin-Isospin Coupling in Nucleons

The total wavefunction of a nucleon pair is the product of its spatial, spin, and isospin parts. With the spatial component being symmetric for L = 0, the combined spin and isospin component must be antisymmetric in order to satisfy the Pauli exclusion principle. This antisymmetry in the spin-isospin part is achieved if the sum of the spin (S) and isospin (I) quantum numbers is an odd integer.

Angular Momentum Addition and Parity

In quantum mechanics, the addition of angular momenta, such as spin and isospin, follows specific rules that determine the symmetry of the resultant state. For a two-nucleon system in which the spatial wavefunction is symmetric (L = 0), the requirement that S + I be an odd integer comes from the need to generate an antisymmetric combination for the remaining degrees of freedom (spin and isospin). This ensures the overall antisymmetry of the fermionic wavefunction.

Pauli Exclusion Principle

This principle states that no two fermions, such as nucleons, can occupy the same quantum state simultaneously. In the context of nucleons, which are fermions, the combined wavefunction must be antisymmetric under the exchange of any two nucleons. This requirement forces a relationship between different parts of the wavefunction: spatial, spin, and isospin.

Spatial Wavefunction Symmetry

When the nucleons are in a state of relative orbital angular momentum L = 0, the spatial part of the wavefunction is symmetric because it has no angular dependence. Since the overall wavefunction must be antisymmetric for identical fermions, the symmetry properties of the spin and isospin parts of the wavefunction must combine to form an antisymmetric state.

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