On the Nature of Energy Levels of Anharmonic Oscillators

Abstract

The physical nature of low-lying energy levels of anharmonic oscillators is shown to be due to quantum reflections on (multiple) complex turning points. Mathematically, certain Stokes' lines linking complex-conjugate turning points may cross the real axis within the classically allowed region at points referred to as theStokes points. The physical significance of the Stokes points lies in the fact that theactualreflection of a quantum particle on a pair of complex-conjugate turning points can alternatively be represented as anequivalent scatteringat the related Stokes point, and so the use of appropriate connection formulae associated with this Stokes point provides an adequate quantitative description of the reflection. Quantization equations for anharmonic oscillators, as derived on making allowance for scatterings at the Stokes points, have the closed form of the Bohr–Sommerfeld formula with an additional term due to the scatterings. Explicit expressions for this additional term are obtained with regard to quartic, sextic, octic, and decadic oscillators. The effect of inactive turning points is discussed. Numerical verification of the quantization equations is performed, and their accuracy is confirmed, with respect to the low-lying energy levels of various anharmonic oscillators, including their ground levels.