Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore.
J Chem Phys. 2023 May 14;158(18). doi: 10.1063/5.0150523.
At the limit of an infinite confinement strength ω, the ground state of a system that comprises two fermions or bosons in harmonic confinement interacting through the Fermi-Huang pseudopotential remains strongly correlated. A detailed analysis of the one-particle description of this "contactium" reveals several peculiarities that are not encountered in conventional model systems (such as the two-electron harmonium atom, ballium, and spherium) involving Coulombic interparticle interactions. First of all, none of the natural orbitals (NOs) {ψn(ω;r)} of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their non-ascendingly ordered occupation numbers {νn} turn out to be related to the eigenfunctions and eigenvalues of a zero-energy Schrödinger equation with an attractive Gaussian potential. This observation enables the derivation of their properties, such as the n-4/3 asymptotic decay of νn at the n→∞ limit (which differs from that of n-8/3 in the Coulombic systems), the independence of the confinement energy vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩ of n, and the n-2/3 asymptotic decay of the respective contribution νntn to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical to those of the two-electron harmonium atom at the ω → ∞ limit, despite the entirely different interparticle interactions in these systems.
在无限约束强度 ω 的极限下，由两个在谐振子约束中相互作用的费米-黄赝势相互作用的费米子或玻色子组成的系统的基态仍然保持强烈关联。对这个“接触体”的单粒子描述的详细分析揭示了一些在涉及库仑相互作用的常规模型系统（如双电子谐振子原子、钡原子和锇原子）中不会遇到的特殊性质。首先，接触体的任何自然轨道（NO）{ψn(ω;r)}都没有被占据，这意味着所有角动量的集体占据都不为零。其次，NO 及其非升序占据数 {νn} 与具有吸引力高斯势的零能薛定谔方程的本征函数和本征值有关。这种观察使我们能够推导它们的性质，例如 νn 在 n→∞ 极限下的 n-4/3 渐近衰减（与库仑系统中的 n-8/3 不同），约束能 vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩与 n 的独立性，以及各自对动能的贡献 νntn 的 n-2/3 渐近衰减。经过适当的缩放，接触体的弱占据 NO 在 ω → ∞ 极限下实际上与双电子谐振子原子的 NO 相同，尽管这些系统中的粒子间相互作用完全不同。
