9 Operator Perturbation Theory for Atomic Systems
165
For the uniform electric field ε(t) = ε. In principle, the more realistic models can
be considered in the framework of our approach. Potential energy in (9.4) has the
barrier. Two turning points for the classical motion along the η axis, t 1 and t 2 , at a
given energy E are the solutions of the quadratic equation (β = β 1 , E = E 0 ):
t 2 =
E
2
0 − 4ε(1 − β)
1/2 − E 0
/ε,
(9.6)
t 1 =
−
E
2
0 − 4ε(1 − β)
1/2 − E 0
/ε, t 1 < t 2 .
(9.7)
Here and below t denotes the argument common for the whole equation system. To
simplify the calculational procedure, the uniform electric field ε in (9.3) and (9.4)
should be substituted by the function [57, 58]:
ε (t) =
1
t
ε
(t − τ )
τ 4
τ 4 + t 4 + τ
(9.8)
with sufficiently large τ (τ = 1.5t 2 ). The function ε(t) practically coincides with
the constant ε in the inner barrier motion region (t < t 2 ) and disappears at t t 2 .
The minimal acceptable value of τ introduced in the spatial dependence of the electric field, which does not influence the final results, can be established experimentally. Thus, the final results do not depend on the parameter τ (the further calculation
has entirely confirmed this fact). Besides the pure technical convenience, the case of
an asymptotically disappearing electric field is more realistic from the physical point
of view. Now we deal with the asymptotically free (without electric field) motion of
the ejected electron along the η-axis. The corresponding effective wavenumber is:
k = (E/2 + ετ/4)
1/2 .
(9.9)
The scattering states energy spectrum now spreads over the range (−ετ/2, +∞),
compared with (−∞, +∞) in the uniform field. In contrast to the case of a free atom
in scattering states in the presence of the uniform electric field remain quantified at
any energy E, i.e. only definite values of β 1 are possible. The latter are determined
by the confinement condition for the motion along the η-axis. The same is true in our
case, but only for E ⊂ (−
1
2 ετ, +
1
2 ετ ). The motion with larger E is non-quantified,
similar to the free atom case.
9.2.2 Energy and Width of the Stark Resonance
The total Hamiltonian H (ς, ν, ϕ) does not possess the bound stationary states. According to OPT [6, 56–58]), one has to define the zero order Hamiltonian H 0 , so
that its spectrum reproduces qualitatively that of the initial one. In contrast to H ,
it must have only stationary states. To calculate the width Γ of the concrete quasistationary state in the lowest PT order one needs only two zeroth-order EF of H 0 :
bound state function Ψ Eb (ε, η, ϕ) and scattering state function Ψ Es (ε, η, ϕ) with the
165
For the uniform electric field ε(t) = ε. In principle, the more realistic models can
be considered in the framework of our approach. Potential energy in (9.4) has the
barrier. Two turning points for the classical motion along the η axis, t 1 and t 2 , at a
given energy E are the solutions of the quadratic equation (β = β 1 , E = E 0 ):
t 2 =
E
2
0 − 4ε(1 − β)
1/2 − E 0
/ε,
(9.6)
t 1 =
−
E
2
0 − 4ε(1 − β)
1/2 − E 0
/ε, t 1 < t 2 .
(9.7)
Here and below t denotes the argument common for the whole equation system. To
simplify the calculational procedure, the uniform electric field ε in (9.3) and (9.4)
should be substituted by the function [57, 58]:
ε (t) =
1
t
ε
(t − τ )
τ 4
τ 4 + t 4 + τ
(9.8)
with sufficiently large τ (τ = 1.5t 2 ). The function ε(t) practically coincides with
the constant ε in the inner barrier motion region (t < t 2 ) and disappears at t t 2 .
The minimal acceptable value of τ introduced in the spatial dependence of the electric field, which does not influence the final results, can be established experimentally. Thus, the final results do not depend on the parameter τ (the further calculation
has entirely confirmed this fact). Besides the pure technical convenience, the case of
an asymptotically disappearing electric field is more realistic from the physical point
of view. Now we deal with the asymptotically free (without electric field) motion of
the ejected electron along the η-axis. The corresponding effective wavenumber is:
k = (E/2 + ετ/4)
1/2 .
(9.9)
The scattering states energy spectrum now spreads over the range (−ετ/2, +∞),
compared with (−∞, +∞) in the uniform field. In contrast to the case of a free atom
in scattering states in the presence of the uniform electric field remain quantified at
any energy E, i.e. only definite values of β 1 are possible. The latter are determined
by the confinement condition for the motion along the η-axis. The same is true in our
case, but only for E ⊂ (−
1
2 ετ, +
1
2 ετ ). The motion with larger E is non-quantified,
similar to the free atom case.
9.2.2 Energy and Width of the Stark Resonance
The total Hamiltonian H (ς, ν, ϕ) does not possess the bound stationary states. According to OPT [6, 56–58]), one has to define the zero order Hamiltonian H 0 , so
that its spectrum reproduces qualitatively that of the initial one. In contrast to H ,
it must have only stationary states. To calculate the width Γ of the concrete quasistationary state in the lowest PT order one needs only two zeroth-order EF of H 0 :
bound state function Ψ Eb (ε, η, ϕ) and scattering state function Ψ Es (ε, η, ϕ) with the