On the potentialx 2N and the correspondence principle

Eigenenergies and frequencies are obtained for a particle oscillating in the potential (1/2)k ^N × ^2N , wherek is a constant,x is displacement, andN is a real number. These potentials include the harmonic oscillator (N = 1) and the square well (N = ). Then ^th eigenenergy has the formA _N n ^(N) , where(N) = 2N/(N + 1), andA _N is independent ofn. Application is made to the correspondence principle for the potentialsN > 1 and it is concluded the classical continuum is not obtained in Bohr's limitn . Complete correspondence is attained in Planck's limith 0, so that for these configurations the limitsh 0 andn are not equivalent. A classical analysis of these potentials is included which reveals the relation log _E (/ _N ) = (N – 1)/2N between frequencyv and energyE, where the constant _N is independent ofE.