Two Nucleons in a Harmonic Oscillator Potential Well

Looking at matrix elements of a two-body interaction we see that simple interactions are functions of r ₁ − r ₂ whereas the wave functions are sums of products of wave functions of r ₁ and of r ₂. In Section 10 it was found useful to expand the interaction in terms of functions of r ₁ and r ₂. It became then a straightforward matter to use methods of tensor algebra to carry out the integrations over d ³ r ₁ and d ³r ₂. Another approach is to make use of the fact that the interaction is a function of the relative coordinate r ₁ − r ₂. This is certainly the case for central interactions V(|r ₁ − r ₂|) with or without spin operators. It is also true for tensor forces (10.71) for which, as shown in (10.72), the potential V_T (|r ₁ − r ₂|) is multiplied by ([s ₁ × s ₂]^(2). [(r₁ − r₂) × (r₁ − r₂)]⁽²⁾). The mutual spin-orbit interaction (10.69) is slightly more complicated but it is expressed in terms of the orbital angular momentum (10.70) which is associated with the relative coordinate. In an alternative approach, the interaction is kept intact and it is attempted to expand the shell model wave functions of two nucleons in terms of the relative coordinate r = r ₁ − r ₂ and the center-of-mass coordinate R = (r ₁ − r ₂)/2. Such an expansion is always possible but in the general case it is rather complicated. It turns out that if the single nucleon wave functions are determined by a harmonic 230oscillator potential well, as described in Section 4, such an expansion can be conveniently carried out. The wave functions discussed in Section 4 do not have the correct asymptotic behavior. Still, they may be good approximations for wave functions of low lying levels in the region where they are large and contribute most to the matrix elements.