As a simple instructive example, let us consider perturbation theory for Stark effect in a hydrogen atom. In a weak field, the ground state energy can be represented as a perturbation expansion in powers of field strength:

Because of possibility of ionization, the ground state has an exponentially small width

and the energy has imaginary part

According to Dyson's argument, the instability of the state leads to the divergence of the perturbation series. As it was established by H.J.Silverstone, the coefficients in the expansion have a factorial growth:

which follows directly from dispersion relations for the energy. Such behaviour is of considerable interest from the general point of view. Moreover, it is useful in the case when the special summation procedure such as Borel method of summation of the divergent series is applied.

Recently, we derive a similar result for relatively new semiclassical approach. In our case, the expansion parameter is the reciprocal of the quantum number, n. The scaled energy is represented in the form of expansion in negative powers of n:

We are considering the Rydberg states with large orbital momentum corresponding in the large n limit to circular orbits of an electron. Such an approach is equivalent to recently developed dimensional expansion. It was shown by D.R.Herschbach with co-workers that this method is well-suited to diverse problems in atomic physics such as atoms in strong fields and two-electron atoms.

We shall treat a hydrogen atom in parallel electric and magnetic fields as a prototype test case. This example has all essential features of the general problem.

We use scaled cylindrical coordinates

The problem reduces to Schrodinger equation with an effective potential

The reciprocal of the quantum number n plays the role of Planck's constant. The scaled field strengths are as follows

The effective potential is nonseparable and tunneling occurs in two coordinates. In a sufficiently small electric field,

the effective potential has a methastable minimum. The typical form of the effective potential is shown in this figure by means of isolevels. At large n, an electron undergoes small oscillations around the minimum. The semiclassical expansion is similar to the methods of molecular vibration analysis. It reduces to the Rayleigh - Schrodinger perturbation theory for anharmonic oscillator.

It was confirmed by direct numerical calculations, that the coefficients in the expansion grow as factorials,

Here we deal with two-dimentional quantum decay problem. In the zero order of quasiclassical theory, the line width is roughly proportional to the exponent

where

is the classical action along the escape path. Our aim is to determine the most probable escape path which minimizes the action integral. Then, the parameter a of the large-order asymptotics is calculated by the formula

Our main result is the development and testing of practical and numerically stable means for calculating the most probable escape path and the action integral. We use two techniques.

The first method is the evaluation of classical trajectories in an inverted potential

by integrating the equations of classical motion

The most probable escape path represents the trajectory beginning on the border of classically accessible region and terminating in the maximum of the inverted potential.

Here, we plot several trajectories, beginning on the zero energy level

By varying the initial coordinates, we choose the single unstable trajectory going along the ridge of inverted potential near the saddle point and approaching its maximum. This trajectory represents the most probable escape path. The action is calculated by the formula

The second method is solving the Hamilton - Jacoby equation for the action:

We expand the potential around the minimum in powers of normal coordinates:

Then, we expand the action

Finally, we sum the expansion. Near the minimum, the lines of constant action represent ellipses. We choose one isoline that contacts the classically accessible region in an escape point. So, we can evaluate the minimum of the action. The results of both methods are in a good agreement.

The dependence of the action on the electric field is shown in the figure. Note, that near the classical ionization threshold, the action tends to zero according to the typical law

In sufficiently strong field, the minimum becomes a complex stationary point and the action becomes a complex number.

Thus, we can determine the divergent behaviour of the semiclassical expansion,

Taking into account this condition, we construct the special variant of Pade - Borel summation procedure. The results of the summation are shown in the table. The first line is the results of Pade - Borel summation without using the asymptotic condition, and the second line is the special variant of Pade - Borel approximants. In both cases, we use the same number of expansion coefficients. The third line is the result of numerical solution. It is evident, that the second method is more precise than the first one.

We call attention to the fact that the problem of quantum tunneling reduces to purely classical equations of motion. This approach invites applications to atoms in external fields and multielectron atoms.

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Designed by A. Sergeev.