Verify the isospin factor √(2) in (12.19). Note that ^14 C, ^14 N^*, ^14 O form an isospin tr

step 1 We need to compute the probability for the hydrogen ground state. So from problem 65, we have p

Verify the isospin factor √(2) in (12.19). Note that ^14 C,...

Verify the isospin factor $\sqrt{2}$ in (12.19). Note that ${ }^{14} \mathrm{C}$, ${ }^{14} \mathrm{~N}^{*},{ }^{14} \mathrm{O}$ form an isospin triplet, which can be viewed as $\mathrm{nn}, \mathrm{np}, \mathrm{pp}$, together with an isospin zero ${ }^{12} \mathrm{C}$ core (see Fig. 2.2). Keep in mind that for indistinguishable proton decays, we must add amplitudes, not probabilities.

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Key Concepts

Isospin Multiplets and Coupling

Isospin multiplets are groups of particles or nuclei that share a common isospin quantum number but differ in their third component. For example, an isospin triplet has three members that can be related by isospin ladder operators. The coupling of isospin states, much like the coupling of angular momentum states, often involves Clebsch–Gordan coefficients, which can lead to factors such as ?2 for normalization and symmetry considerations.

Isospin

Isospin is a symmetry concept in nuclear physics that treats protons and neutrons as two states of the same particle, analogous to how spin describes intrinsic angular momentum. This symmetry simplifies the description of nuclear interactions, particularly when considering the nuclear force which is almost independent of whether a nucleon is a proton or a neutron.

Amplitude Addition for Indistinguishable Particles

When dealing with processes that produce indistinguishable particles, such as identical proton decays, the quantum mechanical amplitudes must be added coherently rather than simply adding the probabilities. This coherent addition can lead to interference effects and factors, like the ?2 factor, that arise from the symmetry properties of the overall wavefunction.

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