Uniformly Valid Asymptotic Approximation for the Quantized Anharmonic Oscillator

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Abstract
A uniformly valid asymptotic expansion is obtained for the solution of the one-dimensional time-independent Schrödinger equation which describes an anharmonic oscillator. The computational method is suggested by McKelvey's treatment of the two-turning-point problem, but has some practical advantages over his procedure. For turning points which lie close together, the energy levels are expressed as an asymptotic power series in ℏ/ (2m)1/2, with coefficients which depend only on the quantum number n and the derivatives of the potential at its minimum. The leading terms of this series are calculated and found to agree with the JWKB and perturbation results of earlier authors. For turning points which do not lie close together, the application of this method is indicated briefly and pursued up to the recovery of the Bohr—Sommerfeld quantum condition.