Quantum theory of anharmonic oscillators-a variational and systematic general approximation method

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Abstract
This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2q2 + lambda q4). As a lowest-order approximation an extremely simple formula for energy levels, Ei(0) = (i + 1/2)1/4(3/alphai + alphai), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alphai is the real positive root of a cubic equation: yi alphai3 + alphai2 - 1 = 0, with yi = 6lambda(2i2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda4 terms, as it should.