Multi-Periodic Coherent States and the WKB-Exactness II ``Non-compact Case and Classical theories Revisited''

On Exactness Of The Supersymmetric WKB Approximation Scheme

Exactness of the lowest order supersymmetric WKB (SWKB) quantization condition $\int^{x_2}_{x_1} \sqrt{E-ω^2(x)} dx = n \hbar π$, for certain potentials, is examined, using complex integration technique. Comparison of the above scheme with a similar, but {\it exact} quantization condition, $\oint_c p(x,E) dx = 2πn \hbar$, originating from the quantum Hamilton-Jacobi formalism reveals that, the locations and the residues of the poles that contribute to these integrals match identically, for both of these cases. As these poles completely determine the eigenvalues in these two cases, the exactnes…

EXACTNESS IN THE WKB APPROXIMATION FOR SOME HOMOGENEOUS SPACES

Analysis of the WKB exactness in some homogeneous spaces is attempted. $CP^N$ as well as its noncompact counterpart $D_{N,1}$ is studied. $U(N+1)$ or U(N,1) based on the Schwinger bosons leads us to $CP^N$ or $D_{N,1}$ path integral expression for the quantity, ${\rm tr} e^{-iHT}$, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.

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.

广义 Langer 修正与所有传统势下 WKB 的精确性

Generalized Langer Correction and the Exactness of WKB for all Conventional Potentials

本文利用 supersymmetric quantum mechanics 形式体系推广了 Langer 修正，证明其能为所有传统势生成精确的 WKB 量子化条件，并揭示了该修正与 SWKB 精确性之间的关联。

Three Dimensional Confinement : WKB Revisited

An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius $r_0$, in the framework of Wentzel-Kramers-Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz and Grabert : The centrifugal term is expanded perturbatively (in powers of $\hbar$), decomposing it into 2 terms -- the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energie…

Classical Aspects of Quantum Walls in One Dimension

We investigate the system of a particle moving on a half line x >= 0 under the general walls at x = 0 that are permitted quantum mechanically. These quantum walls, characterized by a parameter L, are shown to be realized as a limit of regularized potentials. We then study the classical aspects of the quantum walls, by seeking a classical counterpart which admits the same time delay in scattering with the quantum wall, and also by examining the WKB-exactness of the transition kernel based on the regularized potentials. It is shown that no classical counterpart exists for walls with L < 0, and t…