dorsal/arxiv
View SchemaApproximations to path integrals and spectra of quantum systems
| Authors | S. I. Blinnikov, N. V. Nikitin |
|---|---|
| Categories | |
| ArXiv ID | physics/0309060 |
| URL | https://arxiv.org/abs/physics/0309060 |
Abstract
An expression for the Green function G(E;x_1,x_2) of the Schroedinger equation is obtained through the approximations of the path integral by n-fold multiple integrals. The approximations to Re{G(E;x,x)} on the real E-axis have peaks near the values of the energy levels E_{j}. The analytic and numerical examples for one-dimensional and multi-dimensional harmonic and anharmonic oscillators, and Poeschl-Teller potential wells, show that median values of these peaks for approximate G(E;0,0) corresponds with accuracy of order 10% to the exact values of even levels already in the lowest orders of approximation n=1 and n=2, i.e. when the path integral is replaced by a line or double integral. The weights of the peaks approximate the values of the squared modulus of the wave functions at x=0 with the same accuracy.
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"abstract": "An expression for the Green function G(E;x_1,x_2) of the Schroedinger\nequation is obtained through the approximations of the path integral by n-fold\nmultiple integrals. The approximations to Re{G(E;x,x)} on the real E-axis have\npeaks near the values of the energy levels E_{j}. The analytic and numerical\nexamples for one-dimensional and multi-dimensional harmonic and anharmonic\noscillators, and Poeschl-Teller potential wells, show that median values of\nthese peaks for approximate G(E;0,0) corresponds with accuracy of order 10% to\nthe exact values of even levels already in the lowest orders of approximation\nn=1 and n=2, i.e. when the path integral is replaced by a line or double\nintegral. The weights of the peaks approximate the values of the squared\nmodulus of the wave functions at x=0 with the same accuracy.",
"arxiv_id": "physics/0309060",
"authors": [
"S. I. Blinnikov",
"N. V. Nikitin"
],
"categories": [
"physics.comp-ph",
"hep-th",
"physics.atom-ph",
"quant-ph"
],
"title": "Approximations to path integrals and spectra of quantum systems",
"url": "https://arxiv.org/abs/physics/0309060"
},
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