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10.3 Correlations for more general potentials

The result (10.5) can be generalized to any monotonically increasing potentials V such that V ′(r)∕r is decreasing, in the form

⟨ ⟩ m ∑ δ12 ≤ --- V ′(rij) . 8π i<j
(10.7)

The proof is given in Ref. [98], in a polished form due to A. Martin. Some comments are in order:

i) The inequality is saturated in the harmonic-oscillator case. This explains why, with general potentials, the actual value is not too far from the bound, as seen previously in the linear case.

ii) For a power-law potential V = 1 2Brβ with 1 ≤ β ≤ 2, one can get a simple analytic bound on δ12 for the ground state by using the virial theorem and the variational principle. The sequence is

( ) m 3βB ⟨ β ⟩(β−1)∕β m β 3B 1∕β (β−1)∕β δ12 ≤ -------- r12 = ---- --- ⟨VT ⟩ 8π 2( ) [ ]8π 2 ( ) [ ] m β 3B 1∕β 2 (β−1)∕β m β 3B 1∕β 2 (β−1)∕β = ---- --- ------E ≤ ---- --- ------Evar , 8π 2 2 + β 8π 2 2 + β
(10.8)

since the exact energy E is smaller than any variational approximation Evar. A trial wave function of Gaussian shape Ψ(ρ,λ) ∝ exp −1 2α(ρ2 + λ2) leads to

[ β ]2∕(2+β) 3-2-+-β- β- Γ (32-+-2)- Evar = m β 2mg Γ (3) , 2
(10.9)

so finally,

( ) (β−1)∕β ( )1 ∕β [ 3 β ]2 (β−1)∕β(2+β) -β- 3∕(2+β) 6- 3- β-Γ (2-+-2) δ12 ≤ 8π (mg ) β 2 2 Γ (3) . 2
(10.10)

iii) If the potential V grows faster than the harmonic oscillator, there are conflicting contributions from the centrifugal barrier in Eq. (10.4) and from the terms in V ′ whose expectation becomes larger than m∕8π⟨∑ ′ ⟩ V (rij). There are cases where the latter effect dominates. For instance, for V = 1 2Br3, the problem consists of comparing the zero-range correlation with the mean separation. From an accurate hyperspherical calculation, one gets for m = B = 1

9 ⟨ ⟩ δ12 = 0.2263, ---- r212 = 0.2189 . 16π
(10.11)

Hence the inequality (10.7) is clearly violated.

iv) There are in fact good reasons to believe that the inequality (10.7) is inverted if d∕dr(V ′∕r) > 0 everywhere. For perturbations around the harmonic oscillator, i.e., for V = r2 + λw(r), one already has

⟨ ′ 1 ′ 1 ′ ⟩ 1 ⟨ ∑ ′ ⟩ w (r12) − 2ˆr23 ⋅ ˆr12w (r23) − 2ˆr31 ⋅ ˆr12w (r31) > w (rij) 2
(10.12)

at first order, whereas the expectation value of l 2 ρ contributes only at second order. A numerical investigation shows that for V = r2.1 or even V = r2.01, one gets, indeed, the inequality 8πδ12 > m⟨∑ ⟩ V ′(rij).

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