SCALE-INVARIANCE AND THE BOHR-WILSON-SOMMERFELD (BWS) QUANTIZATION FOR POWER-LAW ONE-DIMENSIONAL POTENTIAL WELLS

The classical periods of motion tau(E) are computed for a particle under the influence of a potential well of the form U(x) = alpha Absolute value of x nu, with both nu and alpha positive real constants. Assuming the reflection convention at the origin, these results can be extended to the cases where both nu and alpha 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 tau(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.