POTENTIAL X2N AND THE CORRESPONDENCE PRINCIPLE

Eigenenergies and frequencies are obtained for a particle oscillating in the potential (1/2)kN×2N, where k is a constant, x is displacement, and N is a real number. These potentials include the harmonic oscillator (N = 1) and the square well (N = ∞). The nth eigenenergy has the form ANnλ(N), where λ(N) = 2 N/(N + 1), and AN is independent of n. Application is made to the correspondence principle for the potentials N > 1 and it is concluded the classical continuum is not obtained in Bohr's limit n → ∞. Complete correspondence is attained in Planck's limit h → 0, so that for these configurations the limits h → 0 and n → ∞ are not equivalent. A classical analysis of these potentials is included which reveals the relation logE(ν/νN) = (N - 1)/2 N between frequency v and energy E, where the constant νN is independent of E. © 1979 Plenum Publishing Corporation.