Eigenvalues of an anharmonic oscillator. II

Abstract

An expression is derived for the sixth nonzero term in the WKBJ approximation. The six-term WKBJ approximation is applied to calculate the eigenvalues for the potential V(x) = 1/2 kx2+ax4, k≳0 and a≳0. At low values of λ[ = ah//( μk3)1/2], the calculated results are in excellent agreement, to 15 significant figures, with those of Banerjee et al. for all quantum numbers. At medium and high values of λ, the calculated results are poor at low quantum numbers, but improve rapidly as n increases. A 15-significant-figure accuracy is achieved at n = 10 for λ = 0.05, and at about n = 15 for λ = 20 000. For λ = 0.5, eigenvalues are calculated to 20 significant figures and an argument is presented to show that by n = 50, the calculated value is correct to 19 significant figures. The accuracy further improves at higher quantum numbers.