Quantization conditions

Wang Wen-ze

The quantization energy and the quantization electron orbits of hydrogen atom and Bohr's quantization condition of the steady orbits (or the quantization condition of angular momentum) are educed with the correspondence principle.

Ray LaPierre

Joseph B. Krieger

By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that a necessary condition for the first-order integral to give an exact quantization rule is the vanishing of the higher-order integrals. When this condition is satisfied, the resulting quantization condition is of the form of all previously known exact-quantization rules. The higher-order integrals are shown to vanish for certain cases of interest. An error in Paper I [C. Rosenzweig and J. B. Krieger, J. Math. Phys. 9, 849 (1968)] is noted and an examination of the higher-order integrals shows that a proposed quantization rule given there is not exact.

Nanny Fröman, Per Olof Fröman

Previous work by the present authors on phase-integral quantization conditions for single-well potentials is extended and generalized. Exact as well as approximate quantization conditions are considered and investigated more fully than previously. Not only their necessity but also their sufficiency is treated. An improvement of the estimate of the error of the generalized quantal Bohr–Sommerfeld quantization condition is also given.