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On the four-body problem in the Born–Oppenheimer approximation
Annals of Physics ( IF 3 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.aop.2020.168311
C.A. Escobar , A. Martín-Ruiz

The quantum problem of four particles in $\mathbb{R}^d$ ($d\geq 3$), with arbitrary masses $m_1,m_2,m_3$ and $m_4$, interacting through an harmonic oscillator potential is considered. This model allows exact solvability and a critical analysis of the Born-Oppenheimer approximation. The study is restricted to the ground state level. We pay special attention to the case of two equally heavy masses $m_1=m_2=M$ and two light particles $m_3=m_4=m$. It is shown that the sum of the first two terms of the Puiseux series, in powers of the dimensionless parameter $\sigma=\frac{m}{M}$, of the exact phase $\Phi$ of the wave function $\psi_0=e^{-\Phi}$ and the corresponding ground state energy $E_0$, coincide exactly with the values obtained in the Born-Oppenheimer approximation. A physically relevant rough model of the $H_2$ molecule and of the chemical compound $H_2O_2$ (Hydrogen peroxide) is described in detail. The generalization to an arbitrary number of particles $n$, with $d$ degrees of freedom ($d\geq n-1$), interacting through an harmonic oscillator potential is briefly discussed as well.

中文翻译:

波恩-奥本海默近似中的四体问题

考虑了 $\mathbb{R}^d$ ($d\geq 3$) 中具有任意质量 $m_1,m_2,m_3$ 和 $m_4$ 的四个粒子的量子问题,它们通过谐振子势相互作用。该模型允许精确的可解性和对 Born-Oppenheimer 近似的批判性分析。该研究仅限于基态水平。我们特别注意两个同样重的质量 $m_1=m_2=M$ 和两个轻粒子 $m_3=m_4=m$ 的情况。结果表明,波函数 $\Phi$ 的精确相位 $\Phi$ 的无量纲参数 $\sigma=\frac{m}{M}$ 的幂的 Puiseux 级数的前两项之和psi_0=e^{-\Phi}$ 和相应的基态能量 $E_0$,与在 Born-Oppenheimer 近似中获得的值完全一致。详细描述了 $H_2$ 分子和化学化合物 $H_2O_2$(过氧化氢)的物理相关粗略模型。还简要讨论了对任意数量的粒子 $n$,具有 $d$ 自由度 ($d\geq n-1$),通过谐振子势相互作用的泛化。
更新日期:2020-11-01
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