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SOVIET PHYSICS JETP
APRIL, 1964
VOLUME 18, NUMBER 4
QUANTUM TRANSITIONS TO A CONTINUOUS SPECTRUM, DUE TO ADIABATIC
PERTURBATIONS
A. V. CHAPLIK
Institute of Radiophysics and Electronics, Siberian Division, Academy of Sciences, U.S.S.R.
Submitted to JETP editor May 7, 1963
J. Exptl. Theoret. Phys. (U.S.S.R.) 45, 1518-1522 (November, 1963)
The probabilities for transition from the states of a discrete spectrum to a continuous spectrum under the action of an adiabatic perturbation are computed. The calculations are performed accurate to a numerical coefficient.
QuANTUM transitions between discrete-spectrum states were investigated in the adiabatic approximation by Dykhne [ 1• 2]. It was made clear
that the main feature of the problem is that all the
terms of the adiabatic perturbation theory series
are equal in magnitude, and that the entire series
must be summed to obtain the correct result. This
difficulty is overcome by considering the SchrCidinger equation with complex values of the time. It
is then sufficient to solve the equation in the vicinity of some points of the complex t plane, in
which the adiabaticity conditions wT » 1 are violated ( w -natural frequency of the system and
T -characteristic time of variation of the external conditions). Violation of adiabaticity can be
connected either with singularities of the parameters of the Hamiltonian as functions of the time,
with term crossing, or finally with transition from
the discrete to the continuous spectrum. In the
present paper we consider precisely the last type
of singularity. The characteristic difficulty of this
problem lies in the fact that for the continuous
spectrum the adiabaticity condition wT » 1 is
violated for all instants of time, and not only in
the vicinity of the critical points ( w = 0 for the
continuous spectrum ) .
We shall assume that the potential decreases
sufficiently rapidly with distance, so that the number of negative-energy levels is finite. The case
when terms condense towards the boundary of the
continuous spectrum calls for a separate analysis.
The small parameter of the problem is the quantity a = ( wT) - 1 « 1. The meanings of w and T
are the same as before; w obviously coincides in
order of magnitude with the distance from the last
discrete level to the boundary of the continuous
spectrum E = 0.
The problem is formulated as follows: let the
system be described by a Hamiltonian that depends
parametrically on the time, and is in some state
E 0 of the discrete spectrum as t--oo. It is required to find the probability a(E) of observing the
system in a state with energy E > 0 as t -+oo. This
problem retains the main qualitative singularities
of the ordinary adiabatic situation, such as the like
contribution of all orders of perturbation theory.
There are, however, some essential differences.
It becomes necessary to take into account the virtual transitions between the states of the continuum (see [ 2 ]). For transition to a state with any
energy E, the critical point is the root of the equation E 0 (t) = 0 and not E 0 (t) = E as in the case of
the crossing of terms. Finally, the transition probability contains in the pre -exponential factor a
power-law smallness in the adiabaticity parameter.
1. Let a particle with l = 0 be in a spherically
symmetrical potential U(r ), where the time variation of the parameters of the potential does not
violate the symmetry. We also assume that the
boundary of the continuous spectrum does not
shift, i.e., U(r- oo) = 0 for all t.
We write down the Born-Fock system of equations [3 ], separating explicity the integrations over
the continuous spectrum ( 1i = m = 1 ) :
00
Gn = ~ Knm (t) am (t)
m#'n
+ ~ Kne (t) a (e, t) de,
o
00
a(E) =~KErn (t) am (t)
+ ~ KE. (t) a (e, t)de,
o
m
t
Knm =
~ ~~'!Jmdq exp [ i ~(En- Em) dt'] ,
t
KErn=~ ~~'!Jmdqexp [ iEt- i ~ Emdt'] ,
KE• =~~~'F. dq exp [ i (E- e) t],
(1)
where ¢E, 1/Jn -instantaneous wave functions, satisfying the equations
1046
1047
QUANTUM TRANSITIONS TO A CONTINUOUS SPECTRUM
H1p~) =En (t) 1Vn (t),
H1pE (t)
=
E1VE (t).
(2)
The wave functions of the continuous spectrum
are assumed normalized to an energy a-function.
Equations (1) must be solved in the vicinity of
the point t 0, where any one of the discrete levels
vanishes. The distance from this level to the other
negative-energy levels remains of the same order
of magnitude as that away from t 0 (the terms do
not condense !) . Consequently, in system (1) we
can retain only those terms, which pertain to this
discrete level E 0. As T - - oo we should have
a 0 = 1 and a( E)
0.
Solving these equations with respect to a( E, t),
we get
=
a(E, t)
t
=
KEo [I
+
00
~ d-.~ Ko,a (e, -r) de J
-oo
0
(3)
KE,a (e, t) de.
0
2. Let us clarify the behavior of the matrix elements KoE and KEe: in the region of t close to t 0
of interest to us. The wave function of the state E 0
behaves asymptotically like cr- 1 exp (- k0r ), where
C is a normalization factor and -k3(t)/2 = E 0 (t).
As k0 (t)- 0, the normalization integral diverges
at large r, so that we can calculate C from the
asymptotic expression for lj! 0 (r ). We get
(4)
In the region inside the potential well we can
neglect k3 ( t) as t - t 0 • The Schrodinger equation
assumes the form
X~- 2U (r) Xo = 0,
Xo (r) = 1p 0 (r)!r.
(7)
The dimension of the transition region is determined by the condition
t
~
k2 (-r)
(--T- +E) d-r =
k2 (I
0
t )3
;- o
+ E (t- t
0)
~ 1.
(8)
t,
00
+~
since the critical values of the parameter U0 L 2
are attained at the regular points U0 ( t ) and L ( t )
as functions of the time.
The zeroes of k 0 (t) are located in complexconjugate points of the plane, since k 0 is real on
the real axis. We consider first the simplest case,
when the region of the transition is much smaller
than the distance between the zeroes. The following rep res entation then holds true for k0 ( t ) :
(5)
The potential U ( r) can be written in the form
U0f(r/L ), where U0 and L are the characteristic
depth and width of the well, and the function f(x)
is of the order of unity in the interval x = r /L ~ 1.
By finding for (5) a solution that vanishes at x = 0,
and calculating the logarithmic derivative Xo for
x ~ 1, we obtain a certain function F of a single
parameter U0 L 2 •
The eigenvalue k0 is determined from the equation
(6)
The zeroes of the function F determine the critical
values of the parameter E 0 L 2, for which the discrete levels vanish from the well. We assume
these zeroes to be simple, inasmuch as this is
justified in all the known cases that admit of an
exact solution of the Schrodinger equation. Obviously, the zeroes of k0 (t) will also be simple,
There are two possibilities: either the term in
(8) is much larger than or of the same order as the
second, and then the transition region is t - t 0
~ k02/3 ~ a 113 T and does not depend on E, or else
the second term is much larger than the first. It
is easy to see, however, that we can confine ourselves only to the first case. In fact, this means
that we are considering energies satisfying the
condition E ~ U0a 2/J. On the other hand, it follows from (3) that the amplitude of the probability
of transition into the state E is essentially proportional to exp(iEt 0 ) ~ exp(-ET) ~ exp(E/aU 0 ).
Thus, the main contribution is made by states with
E ~ aU 0 , and we can confine ourselves to a solution of Eq. (3) for E « U0a 213 •
In the transition region E 0 ( t) is of the order of
U0a 213 • It follows therefore that in the calculation
of the matrix elements KoE and KEe: the main
contribution to the integrals will be made by the
regions of values of r whjch are much smaller
than the dimensions of th'~ well. We assume that
the number of discrete levels is of the order of
unity, i.e., U 0 L 2 ~ 1. This means that 1/k0
~ U01/2 a- 113 » L and 1/k 1/ -/2E » uij12a- 113
» L. We must therefore replace the wave functions lJ!o and lJ!E by their asymptotic expressions.
For lJ!o the asymptotic expression is given by (4),
and for lJ!E by
=
(9)
It remains to determine the phase o(k). We see
that the inequalities obtained above, k0L « 1 and
kL « 1, are the conditions for the applicability of
the resonant-scattering theory. In this case, as is
well known (see C4 J), the phase in the asymptotic
expression for the wave function of the continuous
spectrum is given by
A. V. CHAPLIK
1048
tan
o (k)
(10)
= - k I k 0 (t).
We are now in a position, using (4), (9), and (10),
to calculate the coefficients KEo and KEE of (3).
As a result we get
~
•
(k~
K
+ 2£) ''
.
2
1/
_ _!_
ko (2£ · e) '
, p
n (k~ + 2£)';, (k~ + 2f.)"
E• -
+ -2i •~k2o(1:)dTJ,
_1_. i(£-<)l
E- e e
'
=
E (ko (to)r'h,
a (E, t) = v~
't =
'I
.
(t- lo)
[ko Uo)J'\
+ ~ ~ k~ (t) dt) c (W,
exp (iEto
T),
(12)
we obtain the following equation for CfW, T)
ac(W,-r)
a-r
2'1.
-~----'--=-
2';,
X
y:r
Yn (-r2 + 2WJ';,
{I + JrJi: -co~ V
T
---;
't
'
d,;
exp
+ oo),
(14)
and we arrive at the following result:
a (E, t-->
(11)
I,
ko (to)
't __,
+ oo)
1
E' • exp ( iEt 0
g-.-,1
=
(ko) '
1
+ -2i •~' k
2
0
(t)
dt ) . (15)
The energy distribution in the continuous spectrum is of the form
where P is the symbol for the principal value.
The calculation of KEE shows that the integral
in the last term of (3) must be taken in the sense
of the principal value. This question does not
arise with respect to the first term, since the
integrand is continuous.
3. Using (7) and (11), and introducing new variables W and T as well as a new unknown function
C ( W, T ) by means of the formulas
W
= C (W--> 0,
t
lj
2 k0 Vk 0 (2E)·'
KEo = ,r,1 exp [·rEt
r n
g
(·w
+i-1:3)
t
1:
3
t,
2
dn (E)=
dE
{!_ \
ill_ exp
k2 (t) dt} ~l£e-E"
I ko I
2 ·~ o
rc
,
t0
(16)
CJ~2!Imt 0 J.
The total probability of "ionization" P is equal to
P
=
~11: I g
2
1
exp
{f ~ k~ (t) dt}
j/
k0 Ic;'f,- Va e-A/
7 ,
- t~
A-I.
(17)
If we are dealing with atomic collisions 1>, then
the role of the time is assumed by the quantity
jdR/v(R), where R and v are the relative coordinate and the velocity of the nuclei. For the ionization probability as a function of the velocity of
the colliding atoms we obtain the expression
(18)
I
W'
V (2W' + -r'2)3
co-.
~
o
·
)
} 112
iw~
X exp ( - iW',;' -_;_,;' 3 C(W' ,;')dW' +------===
3
'
1t
V2W
+ -r2
-iW'~
xp~Vzw~~-r2~- w,C(W',,;)dW'.
(13)
We were unable to find a solution of Eq. (13). It
is easy to note, however, that this equation does
not contain the parameters of the problem at all.
In addition, we are interested only in the region
W « 1, which corresponds to the inequality presented above E « U0a 213 • We can verify by direct
substitution that for fixed T and as W - 0 the
function C(W, T) tends to a constant. The magnitude of this constant is determined by the behavior
of C ( W, T) in the region W ~ 1, so that we cannot
seek C(W- 0, T) by putting formally W = 0 in
(13).
Thus, the probability of transition to the continuous spectrum is obtained accurate to the numerical factor
where v 0 is a quantity on the order of the orbital
electron velocity.
The author is grateful to A. M. Dykhne and
V. L. Pokrovski1 for valuable remarks and for a
discussion of the work.
1 A.
M. Dykhne, JETP 38, 570 (1960), Soviet
Phys. JETP 11, 411 (1960).
2 A. M. Dykhne, JETP 41, 1324 (1961), Soviet
Phys. JETP 14, 941 (1962).
3 L. Schiff, Quantum Mechanics, McGraw-Hill,
New York, 1949.
4 L. D. Landau and E. M. Lifshitz, Quantum
Mechanics, Pergamon, 1958.
Translated by J. G. Adashko
248
1lFor
example, the decay of negative ions in slow collisions. The number of discrete levels of the "extra" electron
is finite, corresponding to the situation considered above.
1/--pages
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