10.2 The case of linear confinement

Consider first a purely linear potential V (r) = 1 2br. From the positivity of 2 ρ and of ( ₁₂ + ₂₃ + ₃₁)², which ensures that ≥−1 2, when the average is taken for a symmetric wave function, one obtains

(10.5)

For m = b = 1, the bound δ₁₂ ≤ 0.05968 is not too far from the exact values obtained in Chapter 5, which are

(10.6)

Note that δ₁₂ depends slightly on n, unlike in the two-body case. The state with n = 1 contains more contributions from higher hyperspherical harmonics, or, say, from configurations with internal angular momenta l_ρ = l_λ > 0 (still coupled by ρ + λ = 0), and this reduces the value of δ₁₂. For a collective linear potential V _T ∝ ^1∕2, which is exactly hyperscalar, the correlation coefficient δ₁₂ would be strictly independent of n.