费米子系统空间密度振荡的半经典理论
Semiclassical theory for spatial density oscillations in fermionic systems
摘要：
本文研究了束缚在局部（平均场）势$V(\bfr)$中的$N$费米子系统的粒子和动能密度。我们推广了最近发展的半经典理论[J. Roccia和M. Brack，Phys. Rev.\ Lett. { \bf 100}，200408（2008）]，其中根据相应经典系统的闭合轨道计算密度到$D>1$维。我们正则化了在经典转折点附近的$r=0$和$(ii)$上球形系统U（1）对称性破缺的半经典结果$(i)$，在经典转折点附近Friedel振荡是主要的，并且可以通过从$r$到最近转折点的最短轨道很好地再现。对于球对称系统，我们发现存在两种类型的振荡，可以归因于径向和非径向轨道，分别。半经典理论进行测试，对精确的量子力学计算的各种模型的潜力。我们发现一个非常好的整体之间的半经典和精确的数值密度的数值协议，即使是中等粒子数$N$。使用“局部维里定理”，被证明是有效的（除了一个小的区域周围的经典转折点），为任意的本地潜力，我们可以证明，在量子力学密度的振荡再现的振荡部分的$τ_{\text{TF}}[ρ]$ Fermi功能。
英文摘要：
We investigate the particle and kinetic-energy densities for a system of $N$ fermions bound in a local (mean-field) potential $V(\bfr)$. We generalize a recently developed semiclassical theory [J. Roccia and M. Brack, Phys. Rev.\ Lett. {\bf 100}, 200408 (2008)], in which the densities are calculated in terms of the closed orbits of the corresponding classical system, to $D>1$ dimensions. We regularize the semiclassical results $(i)$ for the U(1) symmetry breaking occurring for spherical systems at $r=0$ and $(ii)$ near the classical turning points where the Friedel oscillations are predominant and well reproduced by the shortest orbit going from $r$ to the closest turning point and back. For systems with spherical symmetry, we show that there exist two types of oscillations which can be attributed to radial and non-radial orbits, respectively. The semiclassical theory is tested against exact quantum-mechanical calculations for a variety of model potentials. We find a very good overall numerical agreement between semiclassical and exact numerical densities even for moderate particle numbers $N$. Using a "local virial theorem", shown to be valid (except for a small region around the classical turning points) for arbitrary local potentials, we can prove that the Thomas-Fermi functional $τ_{\text{TF}}[ρ]$ reproduces the oscillations in the quantum-mechanical densities to first order in the oscillating parts.
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