Scale invariance and the Bohr–Wilson–Sommerfeld (BWS) quantization for power law one-dimensional potential wells

Abstract

The classical periods of motion τ(E) are computed for a particle under the influence of a potential well of the form U(x)=α‖x‖ν, with both ν and α positive real constants. Assuming the reflection convention at the origin, these results can be extended to the cases where both ν and α are negative real constants. Also, the scale invariance exhibited by these potentials is analyzed using dimensional arguments directly on the classical equations of motion as well as the more powerful Lie method, appropriate for studying one-parameter symmetry groups of differential equations. The action variables J(E) are obtained from τ(E) and the Bohr–Wilson–Sommerfeld (BWS) quantization rule for the energy spectrum of all the above potentials is reobtained. An interpretation of the results is given in the light of semiclassical arguments.