= Solution
The <deuteron> is the bound <proton>–<neutron> system. Its ground state has $J^P=1^+$ and <isospin> $T=0$, with binding <energy> approximately $2.225\ \mathrm{MeV}$ and no excited bound state. The small binding compared with characteristic nuclear <energy> scales produces an extended tail. With <reduced mass> $\mu\simeq m_N/2$, the asymptotic radial wavefunction decays as $e^{-\kappa r}$, where
$$
\kappa=\frac{\sqrt{2\mu B}}\hbar,
\qquad \kappa^{-1}\simeq4.3\ \mathrm{fm}.
$$
The decay length is not a sharp deuteron <radius> or the force range. It illustrates weak binding outside the region of strong attraction.
The observed total <spin> is consistent with aligned nucleon spins, $S=1$, and a predominantly $L=0$ spatial state, $\,{}^3S_1$. A central attraction identical in both spin–isospin channels would not distinguish this channel from a spin-singlet <proton>–<neutron> channel. In fact the triplet channel binds while the corresponding singlet channel does not. A realistic <nuclear force> must therefore distinguish these spin–isospin channels, for example through a term $V_\sigma(r)\boldsymbol\sigma_p\cdot\boldsymbol\sigma_n$, whose eigenvalues are $+1$ and $-3$ in triplet and singlet states. The absence of a bound dineutron and diproton reinforces the need to distinguish channels; <proton>–<proton> Coulomb repulsion is an additional effect, not a complete explanation of the singlet nuclear interaction.
The clearest evidence against a purely central force is the nonzero <electric quadrupole moment>. A pure isotropic S-wave charge distribution would give zero. The <deuteron quadrupole evidence for a tensor force> instead points to a <nuclear tensor force>,
$$
V_T(r)S_{12},\qquad
S_{12}=3(\boldsymbol\sigma_p\cdot\widehat{\mathbf r})
(\boldsymbol\sigma_n\cdot\widehat{\mathbf r})
-\boldsymbol\sigma_p\cdot\boldsymbol\sigma_n.
$$
It depends on how the spins point relative to the line joining the particles. It preserves total $J$ but couples orbital angular momenta differing by two, producing
$$
|d\rangle=\frac{u(r)}r|{}^3S_1\rangle+\frac{w(r)}r|{}^3D_1\rangle.
$$
Both terms have $J^P=1^+$, and their interference permits a quadrupole moment. The 1939 theoretical analysis explicitly connected the measured quadrupole moment with such directional dependence.<The Electric Quadrupole Moment of the Deuteron and the Field Theory of Nuclear Forces by Christian Møller and Léon Rosenfeld>[Accessible abstract, Nature 144, 476–477]{ref}
The <nuclear magnetic moment> provides a second check. A pure aligned-spin S wave would give approximately $\mu_p+\mu_n\simeq0.880\mu_N$, close to but larger than the observed deuteron value approximately $0.857\mu_N$. The discrepancy is consistent with non-S-wave structure and interaction-current corrections; it is not by itself a model-independent measurement of a D-state probability. The <deuteron D-state admixture> depends on the adopted Hamiltonian and electromagnetic current. The magnetic moments and <mass> differences used for these scales are tabulated physical constants.<Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>[2022 CODATA adjustment, <proton>, <neutron> and deuteron magnetic moments and rest-mass energies]{ref}
A useful schematic interaction therefore contains central, spin-dependent and tensor terms, and possibly <spin>–orbit terms in other channels. Long-range <one-pion exchange> naturally supplies a tensor component, with its finite range set by the pion <mass>. \b[The deuteron establishes an attractive triplet channel and a noncentral tensor interaction; a spin-independent central potential is insufficient.]
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