Weissbach Florian, Pelster Axel, Hamprecht Bodo
Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Sep;66(3 Pt 2A):036129. doi: 10.1103/PhysRevE.66.036129. Epub 2002 Sep 25.
In this paper we introduce a generalization to the algebraic Bender-Wu recursion relation for the eigenvalues and the eigenfunctions of the anharmonic oscillator. We extend this well known formalism to the time-dependent quantum statistical Schrödinger equation, thus obtaining the imaginary-time evolution amplitude by solving a recursive set of ordinary differential equations. This approach enables us to evaluate global and local quantum statistical quantities of the anharmonic oscillator to much higher orders than by evaluating Feynman diagrams. We probe our perturbative results by deriving a perturbative expression for the free energy, which is then subject to variational perturbation theory as developed by Kleinert, yielding convergent results for the free energy for all values of the coupling strength.
在本文中，我们对非简谐振子的本征值和本征函数的代数Bender-Wu递推关系进行了推广。我们将这一著名的形式体系扩展到含时量子统计薛定谔方程，从而通过求解一组递归的常微分方程得到虚时演化振幅。与通过计算费曼图相比，这种方法使我们能够将非简谐振子的全局和局部量子统计量评估到更高阶。我们通过推导自由能的微扰表达式来检验微扰结果，然后将其应用于Kleinert发展的变分微扰理论，从而得到耦合强度所有值下自由能的收敛结果。
