Exactness of Semiclassical Quantization Rule for Broken Supersymmetry

Semiclassical wave equation and exactness of the WKB method

The exactness of the semiclassical method for three-dimensional problems in quantum mechanics is analyzed. The wave equation appropriate in the quasiclassical region is derived. It is shown that application of the standard leading-order WKB quantization condition to this equation reproduces exact energy eigenvalues for all solvable spherically symmetric potentials.

Exactness of SWKB for Shape Invariant Potentials

The supersymmetry based semiclassical method (SWKB) is known to produce exact spectra for conventional shape invariant potentials. In this paper we prove that this exactness follows from their additive shape invariance.

受超对称量子力学启发的半经典方法的最新进展

Recent Advances in Semiclassical Methods Inspired by Supersymmetric Quantum Mechanics

本文回顾了超对称量子力学与半经典方法精确性之间的深刻联系，指出传统形状不变势的数学形式能保证在多种相关情形下产生精确结果，并探讨了这些发现的重要意义。

Exact Semiclassical Evolutions in Relativistic and Nonrelativistic Scalar Quantum Mechanics and Quantum Cosmology

The necessary and sufficient conditions for the exactness of the semiclassical approximation for the solution of the Schrödinger and Klein-Gordon equations are obtained. It is shown that the existence of an exact semiclassical solution of the Schrödinger equation determines both the semiclassical wave function and the interaction potential uniquely up to the choice of the boundary conditions. This result also holds for the Klein-Gordon equation. Its implications for the solution of the Wheeler-DeWitt equation for the FRW scalar field minisuperspace models are discussed. In particular, exact se…

Statistical description of eigenfunctions in chaotic and weakly disordered systems beyond universality

We present a semiclassical approach to eigenfunction statistics in chaotic and weakly disordered quantum systems which goes beyond Random Matrix Theory, supersymmetry techniques, and existing semiclassical methods. The approach is based on a generalization of Berry's Random Wave Model, combined with a consistent semiclassical representation of spatial two-point correlations. We derive closed expressions for arbitrary wavefunction averages in terms of universal coefficients and sums over classical paths, which contain, besides the supersymmetry results, novel oscillatory contributions. Their ph…

A class of exactly solvable models to illustrate supersymmetry and test approximation schemes in quantum mechanics

We derive the analytical eigenvalues and eigenstates of a family of potentials wells with exponential form (FPWEF). We provide a brief summary of the supersymmetry formalism applied to quantum mechanics and illustrate it by producing from the FPWEF another class of exact solutions made of their isospectral partners. Interestingly, a subset of the supersymmetric partners provides a class of exactly solvable double well potentials. We use the exact solutions of the FPWEF to test the robustness and accuracy of different approximation schemes. We determine (i) the ground state through variational …