论文标题
3体谐波分子
3-body harmonic molecule
论文作者
论文摘要
在这项研究中，探索了具有有限休息长度$ r $和零总角动量$ L = 0 $的量子3体谐波系统。它控制三个相同点粒子的近平衡$ s $ s $ s $ s $ s $ s $ s $ψ（r_ {12}，r_ {13}，r_ {23}）$，三个相同点粒子的相互作用，通过任何成对限制势$ v（r_ {12}，r_ {13}，r_ {13}，r_ {13}，r_ {23}，R_ {23}） $ r_ {ij} = | {\ mathbf r} _i- {\ Mathbf r} _j | $之间的粒子之间。在$ r = 0 $的情况下，该系统承认了雅各比坐标中变量的完整分离，它（最大）可整合且可溶解。激发态的整个光谱是退化的，并分析了对相应减少的哈密顿量相关的两个相关的谎言代数表示之间的详细比较。在$ r> 0 $时，问题甚至无法集成或完全可解决，并且退化部分删除。在这种情况下，到目前为止，尚未找到Schrödinger方程的确切解决方案，而其经典对应物被证明是一个混乱的系统。对于$ r> 0 $，使用Lagrange-Mesh方法获得了最低量子状态的总能量$ e $的准确值。在$ 0 $ 0 \ leq r \ leq 4.0 $ 〜A.U的范围内，列出了该州的$ n = 0,1,2,3 $的具体显式结果。 。特别是表明（i）能量曲线$ e = e（r）$开发全球最小值，这是其剩余长度$ r $的函数，并且它在大$ r $的情况下渐近呈有限值，以及（ii）退化状态分为子级别。对于基态，也显示了扰动（小$ r $）和两种参数变分结果（任意$ r $）。简要讨论了该模型与分子物理中应用的扩展。
In this study, the quantum 3-body harmonic system with finite rest length $R$ and zero total angular momentum $L=0$ is explored. It governs the near-equilibrium $S$-states eigenfunctions $ψ(r_{12},r_{13},r_{23})$ of three identical point particles interacting by means of any pairwise confining potential $V(r_{12},r_{13},r_{23})$ that entirely depends on the relative distances $r_{ij}=|{\mathbf r}_i-{\mathbf r}_j|$ between particles. At $R=0$, the system admits a complete separation of variables in Jacobi-coordinates, it is (maximally) superintegrable and exactly-solvable. The whole spectra of excited states is degenerate, and to analyze it a detailed comparison between two relevant Lie-algebraic representations of the corresponding reduced Hamiltonian is carried out. At $R>0$, the problem is not even integrable nor exactly-solvable and the degeneration is partially removed. In this case, no exact solutions of the Schrödinger equation have been found so far whilst its classical counterpart turns out to be a chaotic system. For $R>0$, accurate values for the total energy $E$ of the lowest quantum states are obtained using the Lagrange-mesh method. Concrete explicit results with not less than eleven significant digits for the states $N=0,1,2,3$ are presented in the range $0\leq R \leq 4.0$~a.u. . In particular, it is shown that (I) the energy curve $E=E(R)$ develops a global minimum as a function of the rest length $R$, and it tends asymptotically to a finite value at large $R$, and (II) the degenerate states split into sub-levels. For the ground state, perturbative (small-$R$) and two-parametric variational results (arbitrary $R$) are displayed as well. An extension of the model with applications in molecular physics is briefly discussed.
