Instantaneous Bethe-Salpeter Equation and Its Exact Solution |
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Authors: | CHANG Chao-Hsi CHEN Jiao-Kai LI
Xue-Qian WANG Guo-Li |
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Institution: | 1. CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
;2. Institute of Theoretical Physics, the Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China
;3. Graduate School of the Chinese Academy of Sciences, Beijing 100039, China
;4. Department of Physics, Nankai University, Tianjin 300071, China
;5. Department of Physics, Fujian Normal University, Fuzhou 350007, China |
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Abstract: | We present an approach to solve Bethe-Salpeter (BS) equations
exactly without any approximation if the kernel of the BS
equations exactly is instantaneous, and take positronium as an
example to illustrate the general features of the exact solutions.
The key step for the approach is from the BS equations to derive a
set of coupled and well-determined integration equations in linear
eigenvalue for the components of the BS wave functions
equivalently, which may be solvable numerically under a controlled
accuracy, even though there is no analytic solution. For
positronium, the exact solutions precisely present corrections to
those of the corresponding Schrödinger equation in order v1
(v is the relative velocity) for eigenfunctions, in order v2
for eigenvalues, and the mixing between S and D components in
JPC=1-- states etc., quantitatively. Moreover, we also point out that there is a
questionable step in some existent derivations for the
instantaneous BS equations if one is pursuing the exact solutions.
Finally, we emphasize that one should take the O(v) corrections
emerging in the exact solutions into account accordingly if one is
interested in the relativistic corrections for relevant problems
to the bound states. |
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Keywords: | instantaneous BS equation exact solutions positronium relativistic corrections |
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