谐振势中有限粒子数玻色-爱因斯坦凝聚体的内单粒子密度矩阵
Internal One-Particle Density Matrix for Bose-Einstein Condensates with Finite Number of Particles in a Harmonic Potential
摘要：
研究了在谐振势中有限粒子数（$N$）的玻色-爱因斯坦凝聚体的内部单粒子密度矩阵.我们解决Pethick-Pitaevskii型内部密度矩阵的本征值问题，并找到一个碎片凝聚。相反，凝聚Jacobi型内部密度矩阵给出完全凝聚成单一状态。内部单粒子密度矩阵，因此，示出是不同的，一般为不同的选择的内部坐标系。我们提出了两个物理动机的标准，适当的坐标系的选择，给我们一个唯一的答案，内部的单粒子密度矩阵。一个判据是在无限粒子数极限（$N=\infty$）下，内部单粒子密度矩阵应与实验室坐标系下相应的理想玻色-爱因斯坦凝聚体具有相同的本征值和本征函数。另一个标准是内部单粒子密度矩阵的坐标应该与其余的$(N - 2)$内部坐标正交。这第二个标准是暗示第一个标准。结果表明，内部Jacobi坐标系满足这两个判据，而Pethick和Pitaevskii构造内部单粒子密度矩阵所采用的内部坐标系不满足这两个判据.证明了这两个判据唯一地确定了内部单粒子密度矩阵，它与用雅可比坐标计算的密度矩阵完全相同。这项工作的相关性有关$α$粒子凝聚体的核，以及玻色子原子的陷阱，指出。
英文摘要：
Investigations on the internal one-particle density matrix in the case of Bose-Einstein condensates with a finite number ($N$) of particles in a harmonic potential are performed. We solve the eigenvalue problem of the Pethick-Pitaevskii-type internal density matrix and find a fragmented condensate. On the contrary the condensate Jacobi-type internal density matrix gives complete condensation into a single state. The internal one-particle density matrix is, therefore, shown to be different in general for different choices of the internal coordinate system. We propose two physically motivated criteria for the choice of the adequate coordinate systems which give us a unique answer for the internal one-particle density matrix. One criterion is that in the infinite particle number limit ($N=\infty$) the internal one-particle density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame. The other criterion is that the coordinate of the internal one-particle density matrix should be orthogonal to the remaining $(N - 2)$ internal coordinates. This second criterion is shown to imply the first criterion. It is shown that the internal Jacobi coordinate system satisfies these two criteria while the internal coordinate system adopted by Pethick and Pitaevskii for the construction of the internal one-particle density matrix does not. It is demonstrated that these two criteria uniquely determine the internal one-particle density matrix which is identical to that calculated with the Jacobi coordinates. The relevance of this work concerning $α$-particle condensates in nuclei, as well as bosonic atoms in traps, is pointed out.
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