Zambrano Eduardo, Zapfe W P Karel, Ozorio de Almeida Alfredo M
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany.
Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, R.J., Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Apr;91(4):042911. doi: 10.1103/PhysRevE.91.042911. Epub 2015 Apr 21.
The quantum ergodic conjecture equates the Wigner function for a typical eigenstate of a classically chaotic Hamiltonian with a δ function on the energy shell. This ensures the evaluation of classical ergodic expectations of simple observables, in agreement with Shnirelman's theorem, but this putative Wigner function violates several important requirements. Consequently, we transfer the conjecture to the Fourier transform of the Wigner function, that is, the chord function. We show that all the relevant consequences of the usual conjecture require only information contained within a small (Planck) volume around the origin of the phase space of chords: translations in ordinary phase space. Loci of complete orthogonality between a given eigenstate and its nearby translation are quite elusive for the Wigner function, but our local conjecture stipulates that their pattern should be universal for ergodic eigenstates of the same Hamiltonian lying within a classically narrow energy range. Our findings are supported by numerical evidence in a Hamiltonian exhibiting soft chaos. Heavily scarred eigenstates are remarkable counter-examples of the ergodic universal pattern.
量子遍历猜想将经典混沌哈密顿量典型本征态的维格纳函数等同于能量壳上的δ函数。这确保了简单可观测量的经典遍历期望的计算，与什尼雷尔曼定理一致，但这种假定的维格纳函数违反了几个重要要求。因此，我们将该猜想转移到维格纳函数的傅里叶变换，即弦函数上。我们表明，通常猜想的所有相关结果仅需要弦相空间原点周围小（普朗克）体积内包含的信息：普通相空间中的平移。对于维格纳函数来说，给定本征态与其附近平移之间完全正交的轨迹相当难以捉摸，但我们的局部猜想规定，对于处于经典窄能量范围内的同一哈密顿量的遍历本征态，它们的模式应该是通用的。我们的发现得到了一个表现出软混沌的哈密顿量的数值证据的支持。严重疤痕化的本征态是遍历通用模式的显著反例。
