Eigenvalues of the Potential Function V = z 4 ± Bz 2 and the Effect of Sixth Power Terms

Abstract

The one-dimensional Schrödinger equation in reduced form is solved for the potential function V = z⁴+Bz² where B may be positive or negative. The first 17 eigenvalues are reported for 58 values of B in the range −50⩽B⩽100. The interval of B between the tabulated values is sufficiently small so that the eigenvalues for any B in this range can be found by interpolation. At the limits of the range of B the potential function approaches that of a harmonic oscillator with only small anharmonicity. The effect of a small Cz⁶ term on this potential is studied and it is concluded that a previously reported approximation formula is quite applicable but only for positive values of B. The success of the quartic—harmonic potential function for the analysis of the ring-puckering vibration is shown; it is also demonstrated that the same potential serves as a useful approximation for many other systems, especially those of the double minimum type.