Re-examining the value of old quantization and the Bohr atom approach
The limit of classical orbits approaching for the Coulomb potential (left), the SHO potential (center), and the ISW potential (right). The parameter captures the number of times a path between the maximum radius and the center is traversed in the limit ; for the Coulomb potential this is , for the other two it is .
The numerical energy spectrum of the logarithmic potential plotted with the old quantization spectrum from Eq. (23) . The residual differences between the two are shown in the inset.
The first 50 critical values , plotted with the prediction using old quantization from Eq. (29) . The residual differences between the two are shown in the inset.
The first 20 numerically calculated energy eigenvalues for spherically-symmetric states in the logarithmic potential, along with the first 20 old-quantization energy values. Numerical errors are on the order of , which is generally much smaller than the discrepancy between the numerical eigenvalue and the old-quantization eigenvalue.
The first 20 numerically calculated critical values , together with the old-quantization values. Numerical errors range from to and are in general much smaller than the discrepancy between old-quantization and numerical values.
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