Journal of Modern Physics, 2011, 2, 572-585
doi:10.4236/jmp.2011.26067 Published Online June 2011 (http://www.SciRP.org/journal/jmp)
Problem of Nuclear Decay by Proton Emission in Fully
Quantum Consideration: Calculations of Penetrability and
Role of Boundary Conditions
Sergei P. Maydanyuk, Sergei V. Belchikov
Institute for Nuclear Research, National Academy of Science of Ukraine, Kiev, Ukraine
Received October 9, 2011; revised March 13, 2011; accepted March 29, 2011
Abstract
We develop a new fully quantum method for determination of widths for nuclear decay by proton emission
where multiple internal reflections of wave packet describing tunneling process inside proton-nucleus radial
barrier are taken into account. Exact solutions for amplitudes of wave function, penetrability T and reflection
R (estimated for the first time for decay problem) are found for n-step barrier (at arbitrary n) which approxi-
mates the realistic barrier. In contrast to semiclassical approach and two-potential approach, we establish by
this method essential dependence of the penetrability on the starting point Rform in the internal well where
proton starts to move outside (for example, for the penetrability is changed up to 200 times; accuracy
is ). We impose a new condition: in the beginning of the proton decay the proton starts
to move outside from minimum of the well. Such a condition provides minimal calculated half-life and gives
stable basis for predictions. However, the half-lives calculated by such an approach turn out to be a little
closer to experimental data in comparison with the semiclassical half-lives. Estimated influence of the exter-
nal barrier region is up to 1.5 times for changed penetrability.
157
73 Ta
15
|1|<1.510TR   
Keywords: Tunneling, Multiple Internal Reflections, Wave Packet, Decay by Proton Emission,
Penetrability and Reflection, Half-Life
1. Introduction
Nuclei beyond the proton drip line are ground-state pro-
ton emitters, i.e. nuclei unstable for emission of proton
from the ground state. Associated lifetimes, ranging from
sec to few seconds, are sufficiently long to obtain
wealth of spectroscopic information. Experimentally, a
number of proton emitters has been discovered in the
mass region , 150, and 160 (see [1-4] and ref-
erences in cited papers). A new regions of proton unsta-
ble nuclei is supposed to be explored in close future us-
ing radioactive nuclear beams.
6
10
110A
Initially, the parent nucleus is in quasistationary state,
and the proton decay may be considered as a process
where the proton tunnels through potential barrier. In
theoretical study one can select three prevailing ap-
proaches [5]: approach with distorted wave Born ap-
proximation (DWBA), two-potential approach (TPA),
and approach for description of penetration through the
barrier in terms of one-dimensional semiclassical method
(WKBA). In systematical study these approaches are
correlated between themselves, while calculation of pe-
netrability of the barrier is keystone in successful estima-
tion of gamma widths. While the third approach studies
such a question directly, in the first and second ap-
proaches the penetrability of the barrier is not studied
and the width is based on correlation between wave
functions in the initial state (where the proton occupies
the bound state before decay) and the final one (where
this proton has already penetrated through the barrier
without its possible oscillations inside internal well and it
moves outside). However, the most accurate information
on correspondence between amplitudes and phases of
these wave functions can be obtained from unite picture
of penetration of proton through the barrier, which the
WKBA approach provides (and is practically realized up
to approximation of the second order). Importance of
proper choice of needed boundary condition, the most

S. P. MAYDANYUK ET AL.
573
correctly and closely corresponded to decay, reinforces
our interest in the fully quantum consideration of unite
tunneling process in this task, while the detailed analysis
of selection this boundary condition and its real influence
on results is practically missed in TPA and DWBA ap-
proaches.
Affirmed errors in calculations of half-lives by modern
TPA and WKBA models are about some percents. In this
paper we show that if to take into account influence by
the internal and external regions of the barrier neglected
in TPA, DWBA and WKBA approaches, that one can
obtain change of results up to 200 times (i.e. 20000 per-
cents)! Note that our method has not been accepted by
authors of TPA, DWBA and WKBA models. But it is
easy to clarify effectiveness and proper description and
estimation of the penetration through the barrier in any
model if to use well known tests of quantum mechanics
(like where T and R are penetrability and
reflection concerning the barrier). In this paper we show
that in the WKBA, TPA, DWBA models such tests are
not applicable, while we give apparatus how to work
with them. We analyze in details which approach has
more grounds, is really fully quantum, richer and more
accurate. And we give clear and simple explanation for
difference between our approach and their ones consisted
in essential role of the boundary condition.
=1TR
The main objective of this paper is to pass from semi-
classical unite description of the process of penetration
of proton through the barrier used in the WKBA ap-
proach to its fully quantum analogue, to put a fully
quantum grounds for determination of the penetrability
in this problem. In order to provide such a formalism, we
have improved method of multiple internal reflections
(MIR, see [6-10]) generalizing it on the radial barriers of
arbitrary shapes. In order to realize this difficult im-
provement, we have restricted ourselves by consideration
of the spherical ground-state proton emitters, while nu-
clear deformations are supposed to be further included
by standard way. This advance of the method never stu-
died before allows to describe dynamically a process of
penetration of the proton through the barrier of arbitrary
shape in fully quantum consideration, to calculate pene-
trability and reflection without the semiclassical restric-
tions, to analyze abilities of the semiclassical and other
models on such a basis.
This paper is organized in the following way. In Sec-
tion 2, formalism of the method of multiple internal re-
flections in description of tunneling of proton through
the barrier in proton decay is presented. Here, we give
solutions for amplitudes, define penetrability, width and
half-life. In Section 3, results of calculations are con-
fronted with experimental data and are compared with
semiclassical ones. Here, using the fully quantum basis
of the method, we study a role of the barrier shape in
calculations of widths in details. In particular, for the
first time we observe essential influence of the internal
well before the barrier on the penetrability that necessi-
tates to introduce initial condition which should be im-
posed on the proton decay in its fully quantum consid-
eration. We discuss shortly possible interconnections
between the proposed approach and other fully quantum
methods of calculation of widths. In Section 4, we sum-
marize results. Appendixes include proof of the method
MIR and alternative standard approach of quantum me-
chanics used as test for the method MIR and for the re-
sults presented.
2. Theoretical Approach
An approach for description of one-dimensional motion
of a non-relativistic particle above a barrier on the basis
of multiple internal reflections of stationary waves rela-
tively boundaries has been studied in number of papers
and is known (see [11-13] and references therein). Tun-
neling of the particle under the barrier was described
successfully on the basis of multiple internal reflections
of the wave packets relatively boundaries (approach was
called as method of multiple internal reflections or me-
thod MIR, see [6-9]). In such approach it succeeded in
connecting: 1) continuous transition of solutions for
packets after each reflection, total packets between the
above-barrier motion and the under-barrier tunneling; 2)
coincidence of transmitted and reflected amplitudes of
stationary wave function in each spatial region obtained
by approach MIR with the corresponding amplitudes
obtained by standard method of quantum mechanics; 3)
all non-stationary fluxes in each step, are non-zero that
confirms propagation of packets under the barrier (i.e.
their “tunneling”). In frameworks of such a method,
non-stationary tunneling obtained own interpretation,
allowing to study this process at interesting time moment
or space point. In calculation of phase times this method
turns out to be enough simple and convenient [10]. It has
been adapted for scattering of the particle on nucleus and

-decay in the spherically symmetric approximation
with the simplest radial barriers [6,7,9] and for tunneling
of photons [7,10]. However, further realization of the
MIR approach meets with three questions.
1) Question on effectiveness. The multiple reflections
have been proved for the motion above one rectangular
barrier and for tunneling under it [7,10,13]. However,
after addition of the second step it becomes unclear how
to separate the needed reflected waves from all their va-
riety in calculation of all needed amplitudes. After ob-
taining exact solutions of the stationary amplitudes for
two arbitrary rectangular barriers [6,9], it becomes un-

S. P. MAYDANYUK ET AL.
574
clear how to generalize such approach for barriers with
arbitrary complicate shape. In [14] multiple internal re-
flections of the waves were studied for tunneling through
a number of equal rectangular steps separated on equal
distances. However, the amplitudes were presented for
two such steps only, in approximation when they were
separated on enough large distance, and these solutions
in approach of multiple internal reflections were based of
the amplitudes of total wave function obtained before by
standard method (see Appendix A, Equations (7), (18)
and (19) in this paper). So, we come to a serious unre-
solved problem of realization of th e app roach of mu ltiple
reflections in real quantum systems with complicated
barriers, and clear algorithms of calculation of ampli-
tudes should be constructed.
possible interference between incident and reflected
waves which can be non zero. The penetrability is de-
termined by the barrier shape inside tunneling region,
while internal and external parts do not take influence on
it. The penetrability does not dependent on depth of the
internal well (while the simplest rectangular well and
barrier give another exact result). But, the semiclassical
approach is so prevailing that one can suppose that it has
enough well approximation of the penetrability estimated.
It turns out that if in fully quantum approach to deter-
mine the penetrability through the barrier (constructed on
the basis of realistic potential of interaction between
proton and daughter nucleus) then one can obtain answer
“No”. Fully quantum penetrability is a function of new
additional independent parameters, it can achieve essen-
tial difference from semiclassical one (at the same
boundary condition imposed on the wave function). This
will be demonstrated below.
2) Question on correctness. Whether is interference
between packets formed relatively different boundaries
appeared? Whether does this come to principally differ-
ent results of the approach of multiple internal reflections
and direct methods of quantum mechanics? Note that
such interference cannot be appeared in tunneling through
one rectangular barrier and, therefore, it could not visible
in the previous papers.
2.1. Decay with Radial Barrier Composed from
Arbitrary Number of Rectangular Steps
Let us assume that starting from some time moment be-
fore decay the nucleus could be considered as system
composite from daughter nucleus and fragment emitted.
Its decay is described by a particle with reduced mass m
which moves in radial direction inside a radial potential
with a barrier. We shall be interesting in the radial poten-
tial V(r) with barrier of arbitrary shape which has suc-
cessfully been approximated by finite number N of rec-
tangular steps:
3) Question on unc ertainty in rad ial problem. Calcula-
tions of half-lives of different types of decays based on
the semiclassical approach are prevailing today. For ex-
ample, in [15] agreement between experimental data of

-decay half-lives and ones calculated by theory is
demonstrated in a wide region of nuclei from up
to nucleus with and (see [16] for
some improved approaches). In review [17] methodology
of calculation of half-lives for spontaneous-fission is
presented (see Equations (21-24) in p. 321). Let us con-
sider proton-decay of nucleus where proton penetrates
from the internal region outside with its tunneling
through the barrier. At the same boundary condition,
reflected and incident waves turn out to be defined with
uncertainty. How to determine them? The semiclassical
approach gives such answer: according to theory, in
construction of well known formula for probability we
neglect completely by the seco nd (increa s in g) item of the
wave function inside tunneling region (see [18], Equation
(50.2), p. 221). In result, equality has no
any sense (where T and R are coefficients of penetrability
and reflection). Condition of continuity for the wave
function and for total flux is broken at turning point. So,
we do not find the reflection R. We do not suppose on
106 Te
=1
= 266
d
A=109
d
Z
22
TR

1min1
212
1max
,at <(region1),
,at (region2),
=
,at (region),
NN
VRrr
Vrrr
Vr
VrrR N









 
(1)
where i are constants (). We define the first
region 1 starting from point min , assuming that the
fragment is formed here and then it moves outside. We
shall be interesting in solutions for above barrier energies
while the solution for tunneling could be obtained after
by change ii
V=1i
RN
ki

. A general solution of the wave
function (up to its normalization) has the following form:

 
,,=, ,
lm
r
rY
r


  (2)

11 m1
22
22 12
11
11 21
1m
,at <(region1),
,at (region2),
=
,at (region1),
,at (region),
ik rik r
Rin
ik rik r
ik rik r
NN
nN NN
ik r
N
TNax
eAe Rrr
ee rrr
r
eerrr N
Aerr RN







 












(3)

S. P. MAYDANYUK ET AL.
575
where
j

and
j

are unknown amplitudes, AT and AR
are unknown amplitudes of transmission and reflection,

,
lm
Y



is spherical function,

1
=2
i
kmE
i
V
1
are complex wave number in the corresponding region
with number i, E is energy of the emitted proton. We
shall be looking for solution for such problem by ap-
proach of multiple internal reflections. Here, we restrict
ourselves by a case of the orbital moment while
its non-zero generalization changes the barrier shape
which was used as arbitrary before in development of
formalism MIR and, so, is not principal.
=0l
According to the method of multiple internal reflec-
tions, scattering of the particle on the barrier is consid-
ered on the basis of wave packet consequently by steps
of its propagation relatively to each boundary of the bar-
rier (the most clearly idea of such approach can be un-
derstood in the problem of tunneling through the sim-
plest rectangular barrier, see [7,9,10] and Appendix A
where one can find proof of this fully quantum exactly
solvable method, one can analyze its properties). Each
step in such consideration of propagation of the packet
will be similar to one from the first steps, inde-
pendent between themselves. From analysis of these
steps recurrent relations are found for calculation of un-
known amplitudes for arbitrary step n, summation of
these amplitudes are calculated. We shall be looking for
the unknown amplitudes, requiring wave function and its
derivative to be continuous at each boundary. We shall
consider the coefficients 1, 2, 3 and
2N
TTT
1
R

,
2
R

, 3 R

as additional factors to amplitudes .
Here, bottom index denotes number of the boundary,
upper (top) signs “+” and “–” denote directions of the
wave to the right or to the left, correspondingly. At the
first, we calculate
ikx e
1
T

, 2 T

and 1N T
1
R

, 2 R


1N
R

:
 
1
11
11
22
11
1
11
22
=e,=e
=e,=e.
ik krik kr
jj
,
j
jj jjj
jj
jj jj
ik rikr
jjjj
jjj j
jj
jj jj
kk
TT
kk kk
kkkk
RR
kk kk













(4)
Using recurrent relations:


11
11111 1
=1 1
11
1111 11
=1 1
11 1
=1
=1=
1
=1=
1
=1
mjjj
jjjjjjj j
mjj
mjjj
jjjjjjj j
m
,
,
j
j
jjj jj
m
TRT
RRTRTRRRRR
TRT
RRTRTRR RRR
TTT RR

 
 
 


 

 





 

 




 











 

 

1
1
=,
1
mjj
jj
TT
RR











(5)
and selecting as starting the following values:
111111
=,=, =
NN
RRRRTT
 


,

(6)
we calculate successively coefficients 2N 1
R

R

,
2 1N and
R
R


2
T
1N
T


. At finishing, we de-
termine coefficients
j

:

1
11
=1 1
=1 =
1
mj
jj jj
mjj
T
TRR RR 

 











 ,
1
(7)
the amplitudes of transmission and reflection:
1
=, =
TN R
A
TAR


 
(8)
and corresponding coefficients of penetrability T and
reflection R:
2
1
=,=
n
MIRTMIR R
k
TARA
k
2
. (9)
We check the property:
22
1
=1 or =1,
nTR MIRMIR
kAA TR
k
(10)
which should be the test, whether the method MIR gives
us proper solution for wave function. Now if
the particle is located below then height of one step with
energy of
number m, then for description of transition of this parti-
cle through such barrier with its tunneling it shall need to
use the following change:
.
mm
ki

 (11)
For the potential from two rectangular steps (with dif-
ferent choice of their sizes) after com
all amplitudes obtained by meth
re
parison between the
od of MIR and the cor-
sponding amplitudes obtained by standard approach of
quantum mechanics, we obtain coincidence up to first 15
digits. Increasing of number of steps up to some thou-
sands keeps such accuracy and fulfillment of the prop-
erty (10) (see Appendix B where we present shortly the
standard technique of quantum mechanics applied for the
potential (19) and all obtained amplitudes). This is im-
portant test which confirms reliability of the method
MIR. So, we have obtained full coincidence between all
amplitudes, calculated by method MIR and by standard

S. P. MAYDANYUK ET AL.
576
e define the width of the decay of the studied
mrocedure in [19,23]:
approach of quantum mechanics, and that is way we ge-
neralize the method MIR for description of tunneling of
the particle through potential, consisting from arbitrary
number of rectangular barriers and wells of arbitrary
shape.
2.2. Width  and Half-Life
W
quantum syste following by the p
2
=,
4
p
SF T
m
 (12)
where
p
S
Tis spectroscopic factor and F is normalization factor. is penetrability coefficien
particle from the internal region outside with its tunnel-
ug
t in propagation of the
ing throh the barrier which we shall calculate by ap-
proach MIR or by approach WKB. In approach WKB we
define it so:
 
3
2
2
=exp 2d
R
WKB m
TQVrr




, (13)
2
R


where and are the second and third turning
pointrdin], the normalization f
given bymplifd way as
2
R
s. Acco
si
3
R
g to [15
ie
actor F is
1
F
or by improved way as
2
F
, so:

 
1
2
1
d
=,
Rr
F





1
1
2
2
2
11
2
1π
=dd.
cos 4
R
Rr
RR
kr
Fk
rrr
kr















(14)
The

half-life of the decay is related to the

width by the well known expression:
=ln2.


 (1
For description of interaction betwe
5)
en proton and the
daughter nucleus we shall use th
proton-nucleus potential (at case
ha
nd are Coulomb,
nuclear ponent
e spherical symmetric
=0l) in Ref. [20]
ving the following form:
 
,, =,,
CN l
VrlQv rvrQvr (16)
where

C
vr,

,
N
vrQ a

l
vr
s and centrifugal com

   
2
22
for ,
=
C
r
vr Ze r


2
2
,
3, for <,
2
,, 1
,=, =.
2
1exp
m
m
mm
R
Nl
m
Ze rr
rr
rr
VAZQ ll
vrQ vr
rr mr
d









(17)
Here A and Z are the nucleon and proton numbers of
the daughter nucleus, Q is the Q-value for the pro-
ton-decay,
R
V
of t
he
is the strength of the nuclear component,
R is radius he daughter nucleus,is the effective
radius of t nuclear component, dis diffusen
parameters are defined in [20]. Note that in this paper we
are concentrating on the principal resolution of question
to provide fully quantum basis for calculation of the
penetrability and half-life in the problem of the proton
de
clu
m r
ess. All
cay, while the proton-nucleus potential can be used in
simple form that does not take influence on the reliability
of the developed methodology of the multiple internal
reflections and could be naturally inded for modern
more accurate models.
3. Results
Today, there are a lot of modern methods able to calcu-
late half-lives, which have been studied experimentally
well. So, we have a rich theoretical and experimental
material for analysis. We shall use these nuclei: 157
73 Ta ,
161
75 Re , 167 77 Ir for =0l, and 109 112147r
053 I, 55Cm , 69Tm fo l

. Such a choice we explain by that they
qu
have small
coefficient ofadruple deformation 2

spheri
an
cal.
d at good
We shall a
st
pproximation cansidn be coered as
udy proton-decay on the basis of leaving of the particle
with reduced mass from the internal region outside with
its tunneling through the barrier. This particle is sup-
posed to start from min 1 Rrr

 and move outside ( 1 r
is defined in Equation (1)). Using the coefficients
j
T

and
j
R

in Equations (4) - (6), we calculate total ampli-
tudes of transmission T
A
and reflection
R
A
by Equa-
tion (8), the penetrability coefficient
M
IR
T by Equation
(9). We check the found amplitudes, coefficients
M
IR
T
and
M
IR
R comparing them with corresponding ampli-
tudes and coefficients calculated by the standard ap-
proach of quantum mechanics presented in Appendix B.
We restrict ourselves by Equation (14) for 1
F
and fin
width
d

by Equation (12) and half-live
M
IR

by Equa-
tion (15). We define the penetrability WKB T by Equation
(13), calculate

-width and half-live WKB

by Equa-
tions (12) and (15).
3.1. Dependence of the Penetrability on the
Starting Point
The first interesting result which we have btained is
essential dependence of the penetrability o the position
of the first region where we localize the wave in cidenting
on the barrier. In particular, we have lyzed how
much the internal bo
o
n
ana
undary takes influence on the
la-
equal
min
R
penetrability. In order to obtain well accuracy of calcu
tions, we have chosen width of each interval to be

S. P. MAYDANYUK ET AL.
577
0.01he left boundary of the first
terval as a starting point
fm. We consider tmin R
in
s
tart
gins to move outside and is incident on the internal part
of the barrier in the first stage of the proton decay. In
Figure 1 one can see that half-live of the proton decay of
the 157
73 Ta emitter is changed essentially at displacement
of
R, from proton be- here
s
tart
R. So, we establish essetial dependence of the
penetrability on the starting point
n
s
tart
R, where the pro-
ton starts to move outside by approach MIR. At
= 7.2127
start
R fm this dependence allo to achieve
very close coincidence between the half-live calculated
by the approach MIR and experimental data.
3.2. Dependence of the Penetrability on the
rnal Region
The region of the barrier located ben turning points
2
R and 3 R is main part of the potential used in calcu-
enetrability in the semiclassical approach
(up to the second correction), while the internal and ex-
ternal parts of this potential do not take influ
ws us
e
x
max >RR
e sam
Exte
n of th
us an
s
g width o
twe
nce e
(
be th
latio e p
e
cre
f
in
ence on it.
alyze whether convists in calcula-
in-
da ). Keep-
each interval (st, we shall
L t
a
t
erge
ry max R
ep) to
tions of the penetrability in the approach MIR if to
e the external boun3
ein
increase max R (through increasing number of intervals
he external region), starting from the external turning
point 3 R, and calculate the corresponding penetrability
M
IR
T. In Figure 2 [left panel] one can see how the pene-
trability is changed for 157
73 Ta with increasing max
R.
Dependence of the half-life
M
IR

on max R is shown in
Figure 2 [right panel]. One can see that the method MIR
gives convergent values for the etra half-life
at increasing of max R. From such figures we find that
inclusione external region into calculations changes
the half-life up to 1.5 times (min =0.20
penbility and
of th

sec is the mi-
nimal lf-life calculated at 3max 250RR fm, and
as 0.30
ha
=

sec is the half-life calculated at max =250R
fm, amin error =/1.5
s

  percents). So, error
determination of the penetray in semiclassical
approach (if to take the external region into account) is
expected to be the same as a minimum on such a ba si s.
3.3. Results of Calculations of Half-Lives in Our
and Semiclassical App
e demonstrated above, the fully qu
culatity of the barrier for the proton
decay give us its essential dependence on the starting
point. In order to give power of predictions of half-lives
calculated by the approach MIR, we need to find recip
or 50
bilit
roa
rtainty in calc
in
the
u
ches
As we hav
ions o
to re
antum cal-
e penetrabil
e
e such uncelations of the
f th
bsolva le
half-lives. So, we shall introduce the following hypothesis:
Figure 1. Proton-decay for the nucleus: dependence
of the half-life
157
73 Ta

MIR on the starting point start
R
(at
fm ere calc = 7.2127
start
Rwhulated

MIR
exp
at
th erimentalmax=R250 fm coincides wi exp data

for this nucleus).
Figure 2. Proton-decay for the nucleus: in the left
panel the dependence on penetron the exter-
nal boundary
157
73 Ta
ability 
MIR
max
R
is presente right panel the
dependence half-live d, in the of the

MIR on max
R
is presented (we
use fm whe
= 7.
start
R2127 re calculated

MIR
p
at
fm cowith experimental data max = 250R incides ex

for this nucleus).
In all calculations factor F is the same.

S. P. MAYDANYUK ET AL.
578
we shall assume that in the first stage of the proton decay
proton starts to move outside the most probably at the
coordinate of mini mum of the internal wel l.
If such a point is located in the minimum of the well,
the penetrability turns out to be maximal and half-life
minimal. So, as criterion we could use minimum of
half-live for the given potential, which has stable basis.
We should take into account that the half-lives obtained
before are for the proton occupied ground state while it
needs to take into account probability that this state is
empty in the daughter nucleus. In order to obtain proper
values for the half-lives we should divide them on the
spectroscopic factor S (which we take from [5]), and the
to compare them with experimental data. Results of such
a
omp
ves
oton Decay
fferent ap-
pr
ma-
tion ( given so [5]:
amplitude, which in the distorted wave Born approxi
DWBA) is
1,1;,1
=.
AZA ZApApApA
TV 
 
 (18)
The DWBA calculations of the decay width requires
knowledge of the quasistationary initial state wave func-
tion, , the final state wave function, Ap Ap


e func
1A


, and
inpotential. The initial state wavtion, teraction
1A


, is
ncti
written as a product of the daleus
on,
ughter-nuc
wave fuA

, and the proton wave function, nlj

.
The radial wave function of the proton


=r
l



l

Schröd
should
rr
ing
be i
is d by numerically in
er eqtion with one-body po
rrepart of the Coulomb w
foun
ua
gular
tegrating
tential, and it
ave funct
the
ion,


l
Gr
compl
tion o
inside s
tion in
, in
ex an
f con
pat
wh
n
calculations and experimental dat for some proton
emitters are presented in Table 1. To clete a picture,
we add half-licalculated by the semiclassical ap-
proach to these data.
3.4. Comparison with Other Approaches of
Calculations of Widths of Pr
asymtic region. So, such wav
d itnes non-zero flux. As
tinuf total flux (i.e. absenc
ial r) we cannot obtain zer
ole of its definition, and
particular.
In the final state the wave function of the decaying
nuclear system can be written as a product of the intrin-
sic wave function of the proton and the daughter nucleus
(an inner core). Radial part of the proton wave function
is
pto
defi
ity o
egion
region
e functio
we use co
e of so
o wave fu
at =0r
n is
ndi-
urces
nc-
, in
Half-life of the proton decay is defined on the basis of
width  which can be calculated by di
oaches. For determination of width we shall use sys-
tematics of different approaches proposed in Ref. [5].
The proton emitters are narrow resonances with ex-
tremely small widths. Perturbative approach based on
standard reaction theory could be expected to be accurate.
Let us analyze two following approaches in such a direc-
tion.
3.4.1. The Distorted Wave Born Approximation
Method
The resonance width can be expressed through transition




:
ll
rFrr  , where

l
F
r
rds, this
is the regular Cou-
lomb function. By other wo wave function is real,
and, therefore, it gives zero flux exactly determined on
the basis of the total wave function in the final state. The
total wave functions in the initial and final states corre-
spond to different processes (with different total fluxes).
They, complete wave functions, do not take reflection
from the barrier inside the internal region into account
(but they are defined by different boundary conditions in
the initial and final states only). Here, question about
n emitters. Here, Table 1. Experimental and calculated half-lives of some prototh
p
S
is theoretical spectroscopic factor,

WKB
is half-life calculated by in the semiclassical approach,

MIR is half-life calculate by in the approach MIR, d=


th
pWKB WKB
S
,
=


th
M
IRMIRp
S
, exp

th
p
S
, exp

is experimental data. Values for are used from Table IV in Ref. [5] (see p. 1770); in ca
fm; number of intervals in region from lcula- tions for each nucleus we use: min fm, max =0.11R=R250 min
R
to maximum of the
barrier is 10000, from maximum of the barrier to max
R
is 10000.
Parent nucleus Half-live-values, sec
Nucleus Q, MeV Orbit th
p
S WKB

M
IR

WKB


M
IR


157
73 84
Ta 0.947 1/2 2
s
0.66 1
1.856 10

 1 1.840 10

 1 2.81310

 1 2.789 10  1 3.0 10


161
75 84
Re 1.214 1/2 2
s
0.59 4
1.605 10

 4 1.577 10

 4 2.720 10

 4 2.673 10  4 3.7 10


167
77 90
Ir 1.086 1/2 2
s
0.51 2
2.981 10

 2 2.979 10

 2 5.851 10

 2 5.842 10  2 1.1 10


109
53 56
I 0.829 5/2 1d 0.76 6
2.992 10

 6 3.034 10

 6 3.937 10

 6 3.992 10  4 1.0 10


112
55
5
57
Cm 0.823 0.59
5/2
1d 2.080 10

 5 2.088 10

 5 3.526 10

 5 3.539 10  4 5.0 10


147
69 78
Tm 1.132 0.79
5/2
1d 5
6.250 10

 5 6.159 10

 5 7.911 10

 5 7.796 10  4 3.6 10



S. P. MAYDANYUK ET AL.
579
of the potential (that has
another physical basis for the definition of the
width as definition on the basis of the penet
ba
blem
barrier. E
introd
One can calculate the decay width through time-re-
ato
(or other pote th
trability f a barrie
wel
external region influence on results abso-
lutely (like calcmiclassical aph). But,
thisiblelveproblemccurately and
taking whole stuape of the potial barrier into
account that we demonrateden the fully
quantum approaR.
3.4.2. Thwo-Pal Method
In the mified ttential aproaPA) introduced
by Gurvi and ann i19]ailxam-
ples can be found in [22], see also [5,21,24]) the decay
idth is definede (16) in, ametails):
determination of the decay width is passed on successful
determination of perturbation
decay
rability of the
rrier). However, the question about separation of the
total wave function in the internal region before the bar-
rier into the incident and reflected waves remains unre-
solved in the DWBA method.
Now, if we pass from real radial potential in optical
model approach to complex one, then we shall introduce
new additional independent parameter into our pro
while the penetrability could be calculated for real radial
ssential point in determination of the decay
width in the DWBA method is accurate normalization of
the wave functions in the initial and final states. It could
uce some (essential) uncertainty in calculation of
width also while the penetrability is independent on such
normalization absolutely.
versed capture process. However, in such calculations
shape of the barrier is approximated by inverse oscillr
ntials with knowing exact solutions ofe
wave function) and the peneor suchr
could be calculated. It is clear that both internal l and
do not take
ulations in seproac
s is pos to reso this a
died sh
have
ten
abovst i
ch MI
e Totenti
odwo-popch (T
tzKalbermn [ (dets and e
w so (se [5]nd soe d
 
2
2
4
=d,
nlj l
rB
rW rrr
k



 (19)
where 0 =2kE  ,

is reduced mass,
B
r is ra-
dial coordinate of the barrier height,

nlj r
 is the ra-
dial wave function for the first radial potential including
internal well up to point
B
r,

lr
 is the regular radial
wave function for the second radial potential including
ex ternal region, starting from point
B
r and without the
internal well and with asymptotic behavior
  
0=0,sinπ2 at .
ll l
rkrl r     (20)
Both wave functions are real and defined at different
energy levels. So, in the TPA approach we do not con-
sider fluxes and do not calculate penetrability. We do not
study possible reflection of proton from the barrier in the
internal well for the state which describes the penetration
through the barrier. We escape from a problem of sepa-
ration of the total wave function in the internal into
the inciden
well
t and reflected waves which takes influence
on
ned by increas inter
the resulting penetrability essentially (for example,
for the simplest rectangular barrier with rectangular
well such an uncorrect separation of the same exact
wave function can give infinite penetrability that is ex-
plaied role ofference between inci-
dent and reflected waves). Success in obtaining the re-
sulting width

is dependent on accuracy of corre-
spondence between internal and external wave functions


nlj r
 and


lr
 which should be calculated from
different Schrödinger equations with independent nor-
malization. The correspondence between these wave
functions is determined concerning only one boundary
point
B
r (or it possible shift [24]) separating two poten-
tials and boundary conditions at =0r or at r. In
cont
transm
rincipl
ticular, th
strongly
its shap
rary, the
e o
corresponde
oca
e ex
nce betnt,
itted and reflected wavee MIR approach is
ble or
f lity of quantechanics. r-
e trane ine MIR is
depth intend
rnal wavtion
w
s in th
h th
um m
th
of the
e fun
een the incide
m) that c
In pa
approach
rnal well a
determined concerning the barrier as the whole potential
(with needed restrictions of the radial pro-
responds to fully quantum and unified consideration of
penetration of the proton througe barrier shown in
pnon-l
ile th
smitted wav
pendent on the de
e, whtec


lr
hese de
n the wa

n t
d i
inthe
soluely indent oh
spendence canve
TPA app
and shape
function
roac
(
h b
uc de
is a
h a
t pende
be f
pt
oun


r,
or F
nlj
 but starting fros
directly inclus it alsoher
words, weave strong cpondence een incident,
reflected ad transmitted waves in the MIR approach and
a possible k correspondence betwee
external wave functions in the TPA approach. This pl
the essential role in calcs of the cay widths
ce between the essential de-
m th
de
e implest WKB
). By otapproach fact
horresbetw
n
wee n the internal and
ays
ulationde and
explains so large differen
pendence of penetrability on the starting point in the
MIR approach and practically full absence of such a de-
pendence in the TPA approach.
The simplest example demonstrated why this depend-
ence really exists and it could be not small, can be found
in classical tasks of quantum mechanics. Let us consider
definition of the penetrability in [18] (see Equation
(25.3), p. 103):
2
2
1
=,
k
DA
k (21)
where D is the penetrability, and are wave
umbers of transmitted and inci waves concern-
potential and its as-
1
k
dent 2
k
, i.e.n
ing the left asymptotic part of the
ymptotic right part (see Figure 5 in [18], p. 103), A is the
transmitted amplitude of the wave function. This formula

S. P. MAYDANYUK ET AL.
580
-
ymptotic presentations (2 and (25.2) of waves) and,
so, changes the total penelity D. Result o
tial dependence of the penility of the starting point
he MIR apoa
ic b
ntra
e exact
le examples were analyzed for comparison
o
n her
on (de
flectio
shows that decreasing of the left part of potential in-
creases the wave number 1 k (as is connected with as
5.1)
trabi
etrab
n the essen-
form
R by tprch has the similar sense, but
has been obtained concerning the realstiarrier with
the internal well and takes into account change of the
internal amplitudes also. This codicts with a possible
little dependence of penetrability on the shape of the in-
ternal well in the TPA approach.
Now let us come back to one of the most important
papers at TPA—the paper [22], where somly
. In the solvab
fir
ri
st example with rectangular barrier and well (see Sec-
tion VI.A, pp. 1752-1753, (4.6)) the width does really
not contain dependence on depth of the well, because the
depth is defined as zero initially. Answer on question
what would happen with the width if to displace it below,
the MIR method gives (and penetrability is determined
by Equation (21)). The next example with Coulomb tail
takes already non-zero well into account explicitly. Here,
we already see explicit dependence of the width on the
depth of the well (see Equation (4.18), pp. 1754-1755 in
cited paper), that confirms reliability of logic above. So,
these points seem to be reduction of the TPA approach
and confirm that this approach does not determine the
penetrability in the fully quantum consideration in the
problem of protn decay. At the same time, comparison
of results obtained by such approach and results obtained
by pcipally ot fully quantum developments some-
times leads to some confusion as the TPA approach has
been called as the fully quantum. So, approaches for de-
termination of the decay widths on the basis of penetra-
bility are physically motivated, can be more accurate and
have perspective for research.
4. Conclusions
The new fully quantum method (called as the method of
multiple internal reflections, or MIR) for calculation of
widths for the decay of the nucleus by emission of proton
in the spherically symmetric approximation and the real-
istic radial barrier of arbitrary shape is presented. Note
the following:
 Solutions for amplitudes of wave functiscribed
motion of the proton from the internal region outside
with its tunneling through the barrier), penetrability T
and ren R are found by the method MIR for
n-step radial barrier at arbitrary n. These solutions are
exactly solvable and have been obtained in the fully
quantum approach for the first time. At limit n
these solutions could be considered as exact ones for
the realistic proton-nucleus potential with needed ar-
bitrary barrier and internal hole. Estimated error of
the achieved results is 15 1<1.510TR   .
 In contrast to the semiclassical approach and the TPA
approach, the approach MIR gives essential depend-
ence of the penetrability on the starting point
ce onn of (see Figur
The amplitudes calculated by MIR approac
compared with thesp
tained (for the samential) bependen
form
inside the internal well where proton starts to move
outside in the beginning of the proton decay. For
example, the penetrability of the barrier calculated by
MIR approach for 157
73 Ta is changed up to 200 times
in dependen
R
e 1).
h we
t stan-
positio
corre
e pot
ch
form
R
on
y ind
ding amplitudes ob-
dard stationary approach of quantum mechanics pre-
sented in Appendix B and we obtained coincidence
up to first 15 digits for all considered amplitudes.
This important test confirms that presence of the es-
sential dependence of the p enetrability of the starting
point form
R is result independent on the fully quan-
tum method applied. Su a result raises necessity to
introduce initial condition which should be imposed
on the proton decay in its fully quantum consideration.
Comparison with the WKB and TPA approaches
shows that such approaches have no such a perspec-
tive (having physical sense and opening a possibility
to obtain a new information about the proton decay),
which fully quantum study of the penetrability gives.
 In order to resolve uncertainty in calculations of the
half-lives caused by the dependence of the penetrabil-
ity on form R, we introduce the following initial con-
dition: in the first stage of the proton decay the pro-
ton starts to move outside at the coordinate of mini-
mum of the internal well. Such condition provides
minimal value for the calculated half-life and gives
stable basis for predictions in the MIR approach.
However, the half-lives calculated by the MIR ap-
proach turn out to be a little closer to experimental
data in comparison with the half-lives obtained by the
semiclassical approach (see Table 1).
 Taking the external region of the potential after the
barrier into account, half-live calculated by the MIR
approach is changed up to 1.5 times (see Figure 2).
A main advance of the MIR method developed in this
paper is not a new attempt to describe experimental data
of half-lives more accurately than other approaches do
this, but rather this method seems to be the first tools for
estimation of the penetrability of any desirable barrier of
the proton decay in the fully quantum consideration.
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doi:10.1016/j.ppnp.2006.05.001
and E. M. Lifshitz, “Quantum Mechanics,
oretical Physics,” Nauka, Moscow, Vol. 3,
[18] L. D. Landau
Course of The
1989, p. 768.
[19] S. A. Gurvitz and G. Kälbermann, Decay Width and the
Shift of a Quasistationary State, Physical Review Letters,
Vol. 59, No. 3, 1987, pp. 262-265.
doi:10.1103/PhysRevLett.59.262
[20] F. D. Becchetti and G. W. Greenlees, “Nucleon-nucleus
optical-model parameters, A > 40, E < 50 MeV,” Physi-
cal Review, Vol. 182, No. 4, 1969, pp. 1190-1209.
doi:10.1103/PhysRev.182.1190
[21] D. F. Jackson, M. Rhoades-Brown, “Theories of Al-
pha-Decay,” Annals of Physics, Vol. 105
151-186.
, No. 1, 1977, pp.
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[22] S. A. Gurvitz, “Novel Approach to Tunneling Problems,”
Physical Review A, Vol. 38, No. 4, 1988, pp. 1747-1759.
doi:10.1103/PhysRevA.38.1747
[23] B. Buck, A. C. Merchant, S. M. Perez, “Ground State
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[24] S. A. Gurvitz, P. B. Semmes, W. Nazarewicz and T.
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042705. doi:10.1103/PhysRevA.69.042705

S. P. MAYDANYUK ET AL.
582
le through
nsional rect
Appendix A
Tunneling of Packet through One-Dimensional
Rectangular Step
A main idea and formalism of the multiple internal re-
flections can be the most clearly understood in the sim-
plest problem of tunneling of the partic
one-dimensional rectangular barrier in whole axis [6-10].
Let us consider a problem of tunneling of a particle in
positive x-direction through an one-dimean-
gular potential barrier (see Figure 3). Let us label a re-
gion I for <0x, a region II for 0< <
x
a and a region
III for >
x
a, accordingly. We shall study an evolution
of its tunneling through the barrier.
In standard approach, with energy less than the barrier
height the tunneling evolution of the particle is described
using a non-stationary propagation of wave packet (WP)


0
/

,=,ed,
iEt
x
tgEEkx E 


 (22)
where stationary wave function (WF) is:


=ee,for 0<<,
e,for >
xx
ikx
T
ee,for <0,
ikx ikx
R
Ax
x
xa
Ax







a
(23)
and


1
=2kmE
,

1
1
=2mV E  
, E and m are the
total energy and mass of the particle, accordingly. The
weight amplitude

g
EE
an
can be written in the stan-
dard gaussian formd satisfies to a requirement of the
normalization

2d=1gEEE

he particle. On
, value E is an aver-
age energy of te can calculate coefficients
T
A
,
R
A
,

and

WF
analytically, using a requirements
ity of of a continu

x

rrier. Sub
incid
part 
the in
and its derivative on each
he bstituting in Equation (22)
ent , transmitted
d by Equa-
itted or re-
flected WP, accordingly.
We assume, that a time, for which the WP tunnels
through the barrier, is enough small. So, the time neces-
sary for a tunneling of proton through a barrier of decay,
is about seconds. We consider, that one can ne-
glect a sg of the WP for this time. And a breadth
of the WP appears essentially more narrow on a com-
parison with a barrier breadth. Considering only
sub-barrier processes, we exclude a component of waves
for above-barrier energies, having included the additional
transformation
boun
tr

da
instead of

,kx
tion (23),
ry of t

or
we
a
the
ve

,kx
reflected
recei

,
inc kx


,kx define
nt, transm
ref
cide
21
10
preadin


1,
g
EEgEEV E  
Figure 3. Tunneling of the particle through one-dimen-
sional rectangular barrier.
where

-function satisfies to the requirement

0,for <0
=



;
1, for >0. 

The method of multiple internal reflections considers
the propagation process of the WP describing a motion
of the particle, sequentially on steps of its penetration in
relation to each boundary of the barrier [11-13]. Using
this method, we find expressions for the transmitted and
reflected WP in relation to the barrier. At the first step
we consider the WP in the region I, which is incident
upon the first (initial) boundary of the barrier. This
package transforms into the WP, transmitted through this
boundary and tunneling further in the region II, and into
the WP, reflected from the boundary and propagating
back in the region I. This we consider, that the WP, tun-
neling in the region II, is not reached the second (final)
boundary of the barrier because of a terminating velocity
of its propagation, and consequently at this step we con-
sider only two regions I and II. Because of physical rea-
sons to construct an expression for this packet, we con-
sider, that its amplitude should decrease in a positive
x-direction. We use only one item

exp
x

 
nd incr
in Equ-
ation (23), throwing the secoeasing item


exp
x


of a finite
(in an opposite case werequirement
ness of the WF for an inbarrier).
In result, in the region II we obtain:
break a
definitely wide


10
1
0
/
(,)=ed ,
for 0<<.
tr xiEt
x
tgEEVE E
x
a


 


(25)
Thus the WF in the barrier region constructed by such
way, is an analytic continuation of a relevant expression
for the WF, corresponding to a similar problem with
above-barrier energies, where as a stationary expression
we select the wave


2
exp ikx,
the first step
propagated to the right.
Let’s consider further. One can write ex-
(24)

S. P. MAYDANYUK ET AL.
583
ncident and the reflected WP in rela-
on to the first boundary as follows
pressions for the i
ti






1
0
d,
inc
/
,= ikx iEt
10
ref 1
0
/
for <0,
,= d,
for <0.
Rikx iEt
x
tgEEVEe E
x
x
tgEEVEAe E
x

 


(26)
A sum of these expressions represents the complete
WF in the region I, which is dependent on a time. Let’s
require, this WF and its d continuously
transforo the WF (25) and its derivative at point
=0x (we assume, that the

 

 
at therivative
m int
weight amplitude

g
EE differs weakly at transmitting and reflecting
of the WP in relation to the barrier boundaries). In result,
we obtain two equations, in which one can pass from the
time
and ob
-dependent WP to the corresponding stationary WF
tain the unknown coefficients 0

and 0
R
A
.
At the second step we consider thWP, tling in
d boundary of
the barrier at point
e unne
the region II and incident upon the secon
=
x
a. It transforms into the WP,
transmitted through this boundary and propagated in the
region III, and into the WP, reflected from the
and tunneled back in the region II. For a determination of
these packets one can use Equation (22) with account
whee as the stationary WF we use:
from the une
boundary
Equation (24), r
 


inc
20
tr
20
ref
,,=, for 0 <<,
, for >,
,r 0 <<.
tr
ikx
x
kxkxex a
kxxa
kxx a




(27)
Here, for forming an expression for the WP reflected
bodary, wselect an increasing part of the
stationary solution

0exp

210
,
, fo
x
T
Ae
e
 




x

 y. Imposing a con-
dition of continuity on the time-dependent WF and its
derivative at poin =
onl
t
x
a, we obtain 2uations,
from wind the unknowns co0
T
new eq
hich we fefficients
A
and
0

.
At the third step the WP, tunneling in the reg II, is
in
cted
(28)
Using a conditions of continuity for the time-dependent
WF and its derivative at point , we obtain the un-
knowns coefficients
ion
cident upon the first boundary of the barrier. Then it
transforms into the WP, transmitted through this bound-
ary and propagated further in the region I, and into the
WP, refle from boundary and tunneled back in the
region II. For a determination of these packets one can
use Equation (22) with account Equation (24), where as
the stationary WF we use:


32
inc ref
3
tr
,(,), for 0<<,kxkxxa
kx





2
31
ref
,, for > 0,
,, for 0 <<.
ikx
R
x
Aex
kxex a






=0x
1
R
A
and 1

.
rocesAnalyzing further pblses of the transmis-
sion and the reflection of the WP through the boundaries
of the barrier, we come to a conclusion, that any of fol-
lowing steps can be reduced to one of 2 considered
above. For the unknown coefficients
ossie p
n

, n

,n
T
A
and
n
R
A
, used in expressions for the WP, ft of
e internal reflections from n
obtain the recurrence relations:
orming in r
the boundaries, one ca
esul
som
02
1
=,
nn
ki i k
ik 0
2
=,= e,
=,
=e,=.
nna
R
TR
kik
ki
A
ik ki
AA
ik ik





1
22
nn aikan n
ii






 











(29)
Considering the propagation of the WP by such way,
we obtain expressions for the WF on each region which
can be written through series of multiple WP. Using Eq-
uation (22) with account Equation (24), we determine
resultant expressions for the incident, transmitted and
reflected WP in relation to the barrier, where one can
need to use following expressions for the stationary WF:




inc
tr
=0
ref
=0
,e, for < 0,
,e, for ,
,e, for < 0.
ikx
nikx
T
n
nikx
R
n
kx x
kxAx a
kx Ax




 





(30)
Now we consider the WP formed in result of sequen-
tial n reflections from the boundaries of the barrier and
incident upon one of these boundaries at point =0x
(=1i) or at point =
x
a (=2i). In result, this WP
transforms into the WP

,
i
tr
x
t  , transmitted through
boundary with number i, and into the WP


ref ,
i
x
t  ,
reflected rom this boundary. For an independent on x
parts of the stationary WF one can write:
f
   
  
 
111
ref
=,
tr inc
TR
1
11
222
ref
22
=,
expexp expexp
=,=,
expexp expexp
inc
tr inc2 inc
11 1 1
ref
11
=,
=,
expexp expexp
tr inc inc
x
ikx ikxikx
TR
ikxx xx



 




(or “–”) c -
ne
TR
ikx x xx
  
 


(31)
where the sign “+” orresponds to the WP, tun


ling (or propagating) in a positive (or negative)

S. P. MAYDANYUK ET AL.
584
ecisely describe an
arb ichs formed in result of n-multiple
reflectio to kn a “path” of its propagation along
the barring turrence relations Equation (29),
the coets can be obtained.
x-direction and incident upon the boundary with number
i. Using i T and i R, one can pr
itrary WP wh
ns, if
er. Usi
fficien
ha
ow
he rec
and
i
Ti R
1
0
12 1
1
0
12 1
=, = ,=.
Rnn
RARR

 
Using the recurrence relations, one can find series of
coefficients n
=,= ,=,
nn
TR
nn
nn
AA
TT T 


 
 (32)

, n

, n
T
A
and n
R
A
. However, these
can be calculated easier, using coefficients i T
series

Analyzing all possible “paths” of the WP
propagations along the barrier, we receive:

and
i
R.





2
0
11 21 21
=0 =1
2
0
21
=0 =1
0
21
=0 =1
1= ,
2e
1= ,
2
1=
n
n
R
nn
sub
a
n
n
nn sub
n
n
ni s
kD
ARTRTRRF
ki k
RR F
ki k
RR F





 
 
21 21
=0 =1
4e
1=,
aika
n
n
T
nn sub
ik
ATT RRF



 
 


 
 

 




















,
ub
(33)
where

22
2
222 1
02
2
=.
2,
1,
sub
a
F
kDikD
De


mV
kk 




 
 (34)
 
All series n

, n

, n
T
A

and n
R
A
, obtained
using the method of multiple internal reflections, coin-
cide with the corresponding coefficients

,

, T
A
and
R
A
of the Equation (23), calculated by a stnary atio
methods [18]. Using the following substitution
2,ik

 (35)
where

21
1
=2 kmEV
 is a wave number for a case
of above-barrier energies, expression for the coefficients
n

, n

, n
T
A
and n
R
A
for each step, expressions for
the WF for each step, the total Equations (33) and (34)
the coesponding expressions for a prob-
orticle pgation above this barrier. At the
a of tWP and the time-dep
toge a sign of argument at
tran
lem
tran
one
sform
f th
sform
can
into
e pa
tion
need
rr
ropa
he
chan
endent WF

-function. Besid
22
=0 =0
1.
nn
TR
nn
AA
 

 (36) =
We
f to use only condi-
tion of continuity of the wave function and its derivative
at each boundary, but on the whole region of the studied
potential. At first, we find functions
Appendix B
Direct Method
shall add shortly solution for amplitudes of the wave
function obtained by standard technique of quantum
mechanics which could be obtained i
2
f
and 2
g
(from
the first boundary):

121 21
2
21 1
22
21 12
2
=e,=e
ikkx
ik x
kk k
fg
kk kk



. (37)
Then, using the following recurrent relations:


 
1
2
11 2
12
11
ee,
e
jj
j
j
jj
ik x
jj jjj ik x
jik x
jj jjj
kk fkk
fkk fkk






 (38)
we calculate next functions 3
f
, 4
f
, 5
f
 n
f
, and
by such a formula:

 
1
2ejjj
ikk x
j
12
11
ejj
jj ik x
j
jjjj
gg
kk fkk


  (39)
the functions
k
3
g
, 4
g
, 5
g
 n
g
n
f
and n
g
. From
we find amplitudes n

, n

and amplitude oran
sion
f tsmis-
T
A
:
=0,= =.
n
g
nT
n
A
f
n
  (40)
Now using the recurrent relations:
1111
1 11
1
1
1
eee
ee
jjj j
jj jj
ik xikxikx
jj j
jik xikx
j
g
f


1
jj




 





 (41)
and such a formula:
=,
j
jj j
f
g

  (42)
we consistently calculate the amplitudes , 1n


1n


,
2n


, 2n


 2

, 2

. At finishing, we find ampli-
tude of reflection
R
A
:
 
121121 11
2
22
=eee .
ik k xik k xik x
R
A 

 (43)
As test we use condition:
22
1
=1.
nTR
kAA
k (44)
Studying the problem of proton decay, we used such a
techniques for check the amplitudes obtained previously ellowing property is fulfilled: s the fo

S. P. MAYDANYUK ET AL.
585
nique above. So, result on the large dependence of the
penetrability of the position of the star
in such figures is independent on the used fully quantum
ethod.
by the MIR approach and obtained coincidence up to
first 15 digits for all considered amplitudes. In particular,
we reconstruct completely the pictures of the probability
presented in Figures 1 and 2, but using standard tech-
ting point form R
m