The Hydrogen Atom in Strong Electric Fields: Summation of the Weak Field Series Expansion

The order dependent mapping method, its convergence has recently been proven for the energy eigenvalue of the anharmonic oscillator, is applied to re-sum the standard perturbation series for Stark effect of the hydrogen atom. We perform a numerical experiment up to the fiftieth order of the perturbation expansion. A simple mapping suggested by the analytic structure and the strong field behavior gives an excellent agreement with the exact value for an intermediate range of the electric field, 0.03 ≤ E ≤ 0.25. The imaginary part of the energy (the decay width) as well as the real part of the energy is reproduced from the standard perturbation series. ⋆ e-mail: [email protected] Quite often the perturbation series in quantum theory is not only divergent but even non-Borel summable [1]. Such a non-Borel summability is sometimes known to have its physical origin. For quantum mechanics with an unstable potential or degenerated potential minima the tunneling (barrier penetration) effect gives the Borel singularity [2]. For a uniform electric field in QED, it can explicitly be shown [3] that the instability of the vacuum state causes the non-Borel summability. One may even conjecture [3] that the Borel singularity in QED and QCD is an indication of the instability of the perturbative vacuum. The above observations in turn suggest a possibility that for certain case the standard perturbation series has enough information on the quantum tunneling (or the vacuum instability) that is usually regarded as “non-perturbative” effect. Recently Kleinert [4] has made an interesting observation that the decay width (imaginary part of the energy eigenvalue) for an unstable potential (anharmonic oscillator (AHO) with the negative coupling g < 0) can precisely be reproduced only by using the perturbation series of the energy eigenvalue of AHO, hence explicitly implemented the above possibility. His method is based on the so-called delta expansion [5] (variational perturbation method) that is a special case of the order dependent mapping method (ODM) [6,7]. The convergence of the method for AHO has later been rigorously proven [8] for |g| ≥ g0 (g0 is the radius of convergence of the strong coupling expansion g0 ∼ 0.1). The proof itself [8] is applicable for other quantum systems if one has enough information on the analyticity and the strong coupling behavior of the interested quantity. In this article, we take a simple quantum mechanical system for which the perturbation expansion exhibits the Borel singularity on the positive real axes and apply ODM to re-sum the standard perturbative expansion. The physical origin of the non Borel summability in this system is the instability of the “vacuum.” We consider the ground state energy of the hydrogen atom in a uniform electric