Subatomic Physics

14.11. References 465
14.25. The lowest-lying singlet state of the neutron–proton system, with quantum
numbers J =0,L = 0, is sometimes called the singlet deuteron. It is not
bound, and scattering experiments indicate that it occurs just a few keV
above zero energy; it is just slightly unbound. Assume that the singlet state
occurs at zero energy, and find the relation between well depth and well radius
for a square well. Assume equal singlet and triplet well radii, and show that
the singlet well depth is smaller than the triplet one.
14.26. Show that the tensor operator, Eq. (14.33), vanishes if it is averaged over all
directions ˆr.
14.27. Prove that the operator L = 1
2 (r1 − r2) × (p 1 − p 2) [Eq. (14.37)] is the orbital
angular momentum of the two colliding nucleons in their c.m.
14.28. Show that hermiticity of VNN, Eq. (14.36), demands that the coefficients Vi
be real.
14.29. Show that translational invariance implies that the coefficients Vi in
Eq. (14.36) can depend only on the relative coordinate r = r1 − r2 of the
two colliding nucleons and not on r1 or r2 separately.
14.30. Galilean invariance demands that the transformation
p ′ i = pi + mv
leaves the Vi in Eq. (14.36) unchanged. Show that this condition implies that
Vi can depend only on the relative momentum p = 1
2 (p 1 − p 2).
14.31. Show that the spin operators σ1 and σ2 satisfy the relations
σ 2 x = σ2 y = σ2 z =1
σxσy + σyσx =0
σ 2 =3
(a · σ) 2 = a 2
(σ1 · σ2) 2 =3− 2σ1 · σ2.
14.32. Show that the following eigenvalue equations hold:
σ1 · σ2|t〉 =1|t〉
σ1 · σ2|s〉 = −3|s〉.
Here |s〉 and |t〉 are the spin eigenstates of the two-nucleon system: |s〉 is the
singlet and |t〉 is the triplet state.