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On the potentialx 2N and the correspondence principle

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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 λ(N), whereλ(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 (ν/ν N ) = (N − 1)/2N between frequencyv and energyE, where the constantν N is independent ofE.

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Liboff, R.L. On the potentialx 2N and the correspondence principle. Int J Theor Phys 18, 185–191 (1979).

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  • Field Theory
  • Real Number
  • Elementary Particle
  • Quantum Field Theory
  • Harmonic Oscillator