On Exactness Of The Supersymmetric WKB Approximation Scheme

Details On Exactness Of The Supersymmetric WKB Approximation Scheme On Exactness Of The Supersymmetric WKB Approximation Scheme

1995-12-14

arxiv.org/abs/quant-ph/9512019v1

Physics

Quantum Physics

11 pages, latex, 1 figure available on request

Scientific paper

10.1103/PhysRevA.54.951

Exactness of the lowest order supersymmetric WKB (SWKB) quantization condition $\int^{x_2}_{x_1} \sqrt{E-\omega^2(x)} dx = n \hbar \pi$, for certain potentials, is examined, using complex integration technique. Comparison of the above scheme with a similar, but {\it exact} quantization condition, $\oint_c p(x,E) dx = 2\pi n \hbar$, 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 exactness of the SWKB for these potentials is accounted for. Three non-exact cases are also analysed; the origin of this non-exactness is shown to be due the presence of additional singularities in $\sqrt{E-\omega^2(x)}$, like branch cuts in the $x-$plane.

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