Abstract
We derive an exact quantization condition for the energy levels of a particle in a radial potential assumed finite at the origin. This is used to derive corrections to the semiclassical JWKB quantization condition. We further relate the normalization integral of the wavefunction to the energy derivative of wavefunction at origin and use this expression to derive the corrections to the semiclassical JWKB expressions for the wavefunction at origin. An application to upsilon leptonic decay width is also given.
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References
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For smooth potentials the formulae seem to be good even for smalln values. See. [5]. For an illustrative application of (1.2) to the Υ system see Sect. V of the present paper
Cf. L.D. Landau, E.M. Lifshitz: Quantum mechanics nonrelativistic theory. Transl. J.B. Sykes, J.S. Bell. 2nd Chap. VII Sect. 47: Oxford, New York: Pergamon Press 1965
The extra factor 1/4π in (2.14) is the normalization factor for the angular part of the wavefunction
Eq. (2.14) was derived independently by J.S. Bell. Private communication to one of us (J.P.) in 1979
We have assumedU(0)=0; otherwise replaceE byE−U(0) and\(U\left( { \pm \frac{\hbar }{{\sqrt {2mE} }}} \right)by U\left( { \pm \frac{\hbar }{{\sqrt {2m(E - U(0))} }}} \right) - U(0)in(2.21)\). Notice that a JWKB estimate of the correction can be obtained from the knowledge of\(\frac{{dE}}{{dn}}\) by the inversion formula\(r(V) = \frac{{2\hbar }}{{(2m)^{1/2} }}\int\limits_0^v {\frac{{dE}}{{(V - E)^{1/2} }}\left( {\frac{{dE}}{{dn}}} \right)^{ - 1} }\)
See[5]
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J.S. Bell, J. Pasupathy: Z. Phys. C—Particles and Fields2, 183 (1979)
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Pasupathy, J., Singh, V. Higher order JWKB expressions for the energy levels and the wavefunction at the origin. Z. Phys. C - Particles and Fields 10, 23–27 (1981). https://doi.org/10.1007/BF01545780
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DOI: https://doi.org/10.1007/BF01545780