Application of a novel interpolative perturbation scheme to the determination of anharmonic oscillator wavefunctions

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Abstract
This paper is concerned with a scheme for finding approximate wavefunctions and energy levels of anharmonic oscillators with even power anharmonicities. Generally one can easily find simple asymptotic forms for wavefunctions for small and large oscillator displacements y. In the case of a quartic oscillator the wavefunction with anharmonicity parameter λ, ψ0(y) =exp{−[a11(λ) y4+(2λ/9) y6]1/2} (A) with a11(λ) =E20 and E30−(1/4) E0−(1/3) λ=0, (B) while not exact for all y, does have the correct asymptotic properties. Furthermore, the appropriate root of the cubic equation for E0 deviates by no more than 4% from the exact values of E0 in the range 0⩽λ<∞. In our interpolative method of improving ψ0(y), the correct asymptotic behavior of ψ0(y) is preserved by introducing an extra function of λ into the exponential function (A) and changing the power of the polynomial in the exponent: ψ0(y) =exp{−[a21(λ) y8+a22(λ) y10 +(2λ/9)2y12]1/4}. (C) Through the proper choice of a21(λ) and a22(λ) one obtains ground state energies which deviate by no more than 0.5% over the full range 0⩽λ<∞. Futher improvement is achieved by increasing the degree of the polynomial in the exponential. Excited state wavefunctions are obtained by multiplying exponentials such as (A) and (C) by polynomials in y.