Quantization of quasi-stationary states for the Schrodinger equation: Stark task
We consider a new principle for quantization of the quasi-stationary states for Schrodinger equation with singular potential (the Stark task) and develop a numerical procedure for its realization. The principal tasks, where everybody needs a correct realization of the quasi-stationary and continuum states of the Schrodinger equation, are the scattering and collision problem, Stark task, etc. The standard procedure relates complex eigen-energies (EE) E = Er+0,5iG and complex eigen-functions (EF) to resonances in the spectrum of eigen-values of the Schrodinger equation. The calculation difficulties in the standard methods are well known. The WKB approximation overcomes these difficulties for the states, lying far from "new continuum" boundary and, as rule, is applied in the case of a relatively weak field. Quite another calculation procedures are used in the Borel summation of the divergent perturbation theory (PT) series and in numerical solution of the difference equations following from expansion of the wave-function over finite basis. We developed a consistent approach to the non-stationary state problem solution including also scattering problems (operator perturbation theory). The essence of the method is the inclusion of the well-known method of "distorted waves approximation" in
the frame of the formally exact PT . The zeroth order Hamiltonian H0 of this PT possesses only stationary bound and scattering states. To overcome formal difficulties, the zeroth order Hamiltonian was defined by the set of the orthogonal EF and EE without specifying the explicit form of the corresponding zeroth order potential. The Shrodinger equation for the wave function with taking into account the uniform electric field has the standard form. After separation of variables in parabolic co-ordinates it transformed to the system of two equations for the functions f, g, coupled through the constraint on the separation constants: b1+b2=1. Potential energy has the barrier and two turning points for the classical motion. Further we substitute the external field by the model 1-parameter function, which satisfies to necessary asymptotic conditions. The final results do not depend on parameter of the function. To calculate the width G of the concrete quasi-stationary state in the lowest PT order it is necessary to know two zeroth order EF of H0: bound state function and scattering state function with the same EE. First, one has to define the EE of the expected bound state. It is the well-known problem of states quantization in the case of the penetrable barrier. We solve the system with total Hamiltonian H under the two conditions. These two conditions quantify the bound energy E, separation constant b1. The further procedure for this 2D eigen-value problem is resulted in solving the system of the ordinary differential equations with probe pairs of E, b1. The bound state EE, eigen-value b1 and EF for the zero order Hamiltonian H0 coincide with those for the total Hamiltonian H at field strength approaching to 0. The scattering state functions must be orthogonal to the above-defined bound state function and to each other. It may be written as: g(t)=g1(t)-zg2(t) with g1 satisfying the starting differential equation; the function g2 satisfies the non-homogeneous differential equation, which differs from starting one only by the right hand term, disappearing at t approaching to infiniteness. The
coefficient z ensures the orthogonality condition. The imaginary energy part and resonance width G in lowest PT order are connected directly. The calculation procedure at known energy E and parameter b has been reduced to solution of system of the ordinary differential equations.